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LIBRARY 

OF  THE 

University  of  California. 

Keceived         (•A^cJLr  ■  /^9  7  •  '^ 

Accession  No.  ^  ^/J  Q  J"^  .    Class  No. 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archivaorg/details/elementaryarithmOOfrenrich 


FRENCH'S   MATHEMATICAL    SERIES. 


iLEMENTARY 


JkElSHffilSl© 


FOR    THE    SLATE; 


IN  WHICH 


METHODS    AND    EULES 


ARE  BASED  UPON 


PRINCIPLES  ESTABLISHED  BY  INDUCTION. 


BY 


JOHN  H.  FKEl^CH,  LL.D. 


If  Principles  are'  uunderstood,,  I?f*2fes^«7'a.  useless. 

HARPER     &     BROTHERS. 
1869. 


FRENCH'S  ARITHMETICS. 

This  Series   consists  of  Five    Books,    viz.: 
I.—FIUST    I.JSSSOKS    JJ^    KTTMBFiltS. 
II.  —  EJLEMENTjLMT   jLMITMMETIC. 

III.  —  M  ENTA Ij    a. MITSMETIC In  Press. 

IV.  — COMMON    SCHOOL    AMITMM ETIC. 
v.  — ACADEMIC    AMITSMETIC In  Preparation. 


Tlie  Publlsliers  present  this  Series  of  Text-Books  to  American 
Teacliers,  fully  believing  tliat  tliey  contain  many  new  and  valu- 
able features  tliat  will  especially  commend  them  to  the  practical 
wants  of  the  age. 

The  plan  for  the  Series,  and  for  each  book  embraced  in  it,  was 
fully  matured  before  any  one  of  the  Series  was  completed ;  and 
as  it  is  based  upon  true  philosophical  principles,  there  is  a  har- 
mony, a  fitness,  and  a  real  progressiveness  in  the  books  that  are 
not  found  in  any  other  Series  of  Arithmetics  published. 


Entered,  according  to  Act  of  Congress,  in  the  year  1867,  by 

HAEPER    &    BEOTHERS, 

In  the  Clerk's  Office  of  tlie  District  Court  of  the  United  States  for  the  Southern 

District  of  New  York. 


PREFiiCB, 


THE  object  of  this  book,  designed  especially  for  beginners  in 
Written  Arithmetic,  is  twofold,  viz. :  1st.  To  give  to  young 
learners  a  good  foundation  for  the  study  of  the  Science  of  Numbers, 
by  basing  all  Methods  of    Operation  upon  Principles ;  and,  2d.  To 

f^ive  them  as  much  knowledge  as  possible  of  the  business  affairs  of 
ife,  by  the  introduction  of  business  transactions  stated  in  Correct 
business  language. 

The  plan  of  the  work  differs,  in  most  of  its  essential  points,  from 
that  of  other  works  of  a  like  grade.  To  these  points  of  difference — 
and  it  is  confidently  believed  of  superiority — the  attention  of  parents 
and  teachers  is  particularly  invited. 

General  Divisions. — Chapters. — The  work  is  divided  into  six 
chapters,  the  first  one  of  which  is  devoted  to  Integers ;  the  second, 
to  Decimals  ;  the  third,  to  Compound  Numbers ;  the  fourth,  to  Frac- 
tions; the  fifth,  to  Percentage,  and  the  sixth,  to  Miscellaneous 
Review  Problems. 

Integers  and  Decimals  are  but  parts  of  the  same  class  of  numbers, 
the  latter  being  an  extension  of  the  decimal  scale  to  the  right  of  the 
decimal  point,  or  below  ones.  They  are  both  subject  to  the  same 
laws,  and  all  operations  upon  them  are  based  upon  the  same  princi- 
ples. Therefore,  in  the  natural  order  of  arrangement  of  subjects,  the 
proper  place  for  Decimals  is  immediately  after  Integers. 

Compound  Numbers  differ  from  Integers  and  Decimals  only  in  the 
scales  of  increase  and  decrease,  which,  in  the  latter,  are  uniform  and 
decimal,  while  in  the  former,  they  are  irregular  and  varying.  The 
new  facts  to  be  learned  in  Compound  Numbers  are,  the  scales  or 
tables,  and  their  application  to  the  processes  of  Addition,  Subtraction, 
Multiplication,  and  Division.  Only  two  of  the  denominations  given 
in  this  book— 5i  or  5.5  yards  are  1  rod,  and  30^  or  30.25  square  yards 
are  1  square  rod— are  Mixed  Numbers,  and  these  are  as  well  expressed 
decimally  as  fractionally.  There  is,  therefore,  no  good  reason  why 
Fractions  should  precede  Compound  Numbers,  no  knowledge  of  the 
former  being  necessary  in  studying  the  latter,  while  the  advantages 
of  the  reverse  order  of  arrangement  are  obvious. 

A  knowledge  of  the  preceding  chapters  prepares  the  pupil,  on 
reaching  Fractions,  to  comprehend  the  new  facts  to  be  learned,  viz. : 
the  Notation  of  Fractions,  the  General  Principles  of  this  class  of 
numbers,  and  the  application  of  these  principles  to  the  operations 
upon  Fractional  Numbers. 

In  the  general  arrangement  of  the  work,  and  also  in  its  details,  the 
fact  has  never  been  lost  sight  of,  that  only  a  small  portion  of  all  the 
children  who  commence  the  study  of  Arithmetic  go  through  their 
text-book ;  and  that  a  child  should  be  taught  first  that  which  it  is  most 
desirable  and  important  for  him  to  know ;  so  that,  whenever  he  leaves 
school,  the  knowledge  he  has  acquired  will  be  of  practical  value  to 
him  in  after  life. 


IV  PREFACE. 

Subdivisions.— Sections  and  Cases. — Each  chapter  except  the 
last  is  divided  into  sectious,  and  wherever  necessary,  the  sections  are 
subdivided  into  cases.  The  subjects  of  corresponding  sections  in  the 
first  four  chapters  are  similar.  For  example,  Section  I.  in  each  chap- 
ter is  Notation  and  Numeration ;  Section  II.  is  Addition,  etc.  The 
cases  in  the  several  Sections  correspond  to  each  other,  wherever  the 
nature  .of  the  subjects  will  admit.  Especial  attention  is  invited  to 
the  following  points  in  the  several  chapters  and  sections : 

Chapter  I. — The  first  Method  of  Addition  will  familiarize  pupils 
with  the  reason  for  the  "  carrying  process,"  and  also  accustom  them 
to  add  the  reserved  tens  of  the  sum  of  any  column  to  the  first  figure 
of  the  next  column,  instead  of  the  last. 

All  the  cases  in  Multiplication  and  Division  are  based  upon  a  few 
general  principles,  readily  understood,  and  hence  easily  remembered. 

Long  Division  precedes  Short  Division,  because  it  is  simpler.  In 
the  former,  all  the  partial  results — quotient  figure,  partial  dividend, 
partial  product,  and  partial  remainder— are  written  down  ;  while  in 
the  latter,  the  quotient  figure  only  is  written,  the  memory  being  taxed 
to  form  all  the  combinations,  and  retain  all  the  other  partial  results 
in  the  process.  Long  Division  is  a  general  process,  while  Short 
Division  is  a  contraction,  limited  in  its  application.  The  partial 
results  written  in  the  former  are  really  Visible  Objects,  while  in  the 
latter  they  become  Abstract  Numbers.  The  natural  order  of  mental 
development,  Perception  before  Memory,  has  therefore  been  observed, 
in  placing  Long  Division  before  Short  Division. 

The  divisor  is  written  at  the  right  of  the  dividend.  This  arrange- 
ment is  as  convenient  for  Short  Division  as  that  of  placing  the  divisor 
at  the  left  of  the  dividend ;  while  in  Long  Division,  the  quotient  is 
written  under  the  divisor,  and  the  factors  of  the  partial  dividends  are 
thus  brought  nearer  together,  and  therefore  in  a  more  convenient 
position  for  multiplication. 

A  section  embracing  the  simplest  cases  in  Measurement  is  intro- 
duced into  this  chapter,  because,  1st.  The  subject  is  interesting  to 
children,  and  is  readily  understood  by  them  as  soon  as  they  have 
passed  over  the  fundamental  rules;  and,  2d.  The  cases  here  given  are 
the  basis  of  the  objective  method  used  in  illustrating  some  of  the 
principles  of  Decimals. 

Chapter  II. — The  Diagram  of  Decimal  Notation,  the  Table  of 
Values  of  Decimal  Numbers,  and  the  Decimal  Notation  and  Nume- 
ration Table,  when  thoroughly  understood,  give  to  pupils  a  clearer 
comprehension  of  Decimals  than  they  can  obtain  without  these  aids. 

The  reason  for  placing  the  decimal  point  in  the  product  in  Multi- 
plication is  derived  from  the  principles  of  Measurement ;  and  that 
for  placing  the  decimal  point  in  Division,  from  a  general  principle  of 
Division.  These  reasons  are  strictly  philosophical,  and  easily  under- 
stood, and  are  entirely  independent  of  Fractions. 

The  divisions  of  the  dollar  being  decimal,  and  all  computations  in 
U.  S.  money  being  based  upon  the  same  principles  as  f)ecimals,  the 
subject  of  U.  S.  or  Federal  Money  is  embraced  in  this  chapter,  and 
the  necessity  for  separate  principles  and  rules  is  obviated. 


PREFACE.  V 

Chapter  III.— The  Tables  of  Compound  Numbers  are  arranged  in 
the  order  in  which  they  will  be  the  most  readily  comprehended  by 
young  pupils ;  and  only  those  denominations  in  actual  use  are  given. 
A  few  tables,  such  as  Troy  Weight,  Surveyors'  Measure,  etc.,  being  of 
limited  use,  are  omitted. 

The  Tables  of  the  Metric  System  are  given,  because  they  are  legal- 
ized by  act  of  Congress  ;  but  they  are  not  made  prominent,  because 
they  are  not  yet  in  use. 

Chapter  IV. — The  term  Similar  Fractions  is  used  in  place  of 
Fractions  having  a  cornmon  denominator.  The  simplicity  and  com- 
prehensiveness of  the  term  should  secure  its  general  adoption.  The 
only  important  application  of  the  subject,  Common  Multiple,  is  in 
the  reduction  of  dissimilar  to  similar  fractions.  It  is  therefore  pre- 
sented in  this  chapter.  The  subjects  of  Least  Common  Multiple,  and 
Common  Divisors,  not  being  essential  to  an  elementary  work,  have 
been  omitted. 

The  cases  in  Multiplication  and  Division  are  different  from  those  in 
any  similar  work,  and  the  methods  are  also  new  and  superior.  The 
method  of  Division  is  based  upon  the  same  general  principle  as  is 
Division  of  Decimals. 

The  applications  of  Cancellation  to  Multiplication  and  Division  are 
made  the  subject  of  a  separate  section. 

Chapter  V. — One  general  case  is  given,  embracing  all  the  general 
principles  of  Percentage;  and  to  this  case  all  the  methods  for  com- 
putations in  Insurance,  Commission,  Profit  and  Loss,  Stocks,  Banking, 
and  Interest,  are  referred. 

The  method  for  Interest  is  new,  and  its  simplicity,  absolute  accu- 
racy, and  general  application,  make  it  superior  to  any  heretofore 
presented. 

Chapter  VI.— The  problems  in  this  chapter  embrace  applications 
of  all  the  principles  and  methods  of  computation  contained  in  the 
previous  chapters  of  the  book. 

Inductions.— Each  chapter*  as  well  as  many  of  the  sections  and 
cases,  commences  with  Illustrations  which  form  Visible  Objects. 
Then  follow,  in  the  natural  order.  Concrete  and  Abstract  Numbers. 

Illustrations.— The  cuts,  maps,  and  diagrams,  all  of  which  are 
new,  are  intended  not  only  to  aid  the  pupil  in  acquiring  a  clear  under- 
standing of  the  subjects  they  illustrate,  but  also  to  educate  his  eye, 
cultivate  his  taste,  and  teach  him  some  useful  fact  or  principle. 

•  Examples  and  Problems. — Care  has  been  taken  to  use  these 
term^ — so  often  used  indiscriminately — in  accordance  with  their  sig- 
nification. 

In  the  induction  to  a  Case  or  Method,  one  or  more  examples  are 
solved,  and  the  solution  is  inductively  explained.  These  examples, 
except  in  Currency,  Compound  Numbers,  and  Percentage,  contain 
only  abstract  numbers,  because  a  general  principle  should  not  be 
deduced  from  a  special  or  limited  application. 


VI  PREFACE. 

The  problems  are  derived  from  actual  business  tranactions,  and  are 
all  new.  The  facts  stated  in  them  have  been  obtained  from  reliable 
authorities,  and  the  business  transactions  are  in  accordance  with 
business  customs. 

Each  chapter  closes  with  a  section  of  Review  Problems,  designed 
to  test  the  pupil's  knowledge  of  all  the  previous  chapters. 

Definitions. — The  definitions,  being  intended  for  young  minds, 
are  stated  in  the  inductive  form.  They  are  brief,  accurate,  and  com- 
prehensive. 

Oral  Exercises.— It  is  a  conceded  fact  that  children  leam 
methods  and  processes  of  computation  more  readily  than  they  learn 
combinations.  Many  persons  go  on  through  life  performing  all  their 
numerical  computations  by  counting.  They  never  learn  to  step  more 
than  a  one  at  once.  The  Oral  Exercises,  if  practiced  according  to  the 
directions  given,  will  break  up  the  counting  plan — pupils  will  learu 
to  step  from  given  parts  to  required  results  without  hesitation,  and 
will  soon  become  rapid  and  accurate  computers. 

Tables  of  Combinations. — The  Addition,  Subtraction,  Multi- 
plication, and  Division  tables  are  presented  in  a  new,  and,  it  is  believed, 
a  more  attractive  form  than  the  solid  pages  of  figures  that  have 
greeted  the  eyes  of  children  from  time  immemorial. 

Principles  and  Rules. — Principles  are  deduced  immediately 
from  the  inductive  examples,  and  are  followed  by  problems  which 
require  the  pupil  to  apply  the  principles.  He  is  thus  made  familiar 
with  reasons  for  the  processes,  before  the  rules  are  given ;  and  whenever 
he  applies  a  rule  in  solving  a  problem,  the  why  is  as  familiar  to  him 
as  the  how.  Rules  without  principles  are  soon  forgotten ;  while  if 
principles  are  understood,  rules  are  useless. 

Methods  of  Proof. — Self-reliance  is  one  of  the  most  important 
things  that  can  be  taught  to  children.  To  do  their  work  correctly, 
and  to  feel  sure  that  they  are  correct,  they  must  be  drilled  in  combi- 
nations until  they  add,  subtract,  multiply,  and  divide  without  making 
any  mistakes.  As  methods  of  proof  generally  retard  children  in 
reaching  this  most  desirable  degree  of  accuracy,  they  have  been 
omitted  from  this  work. 

Teachers'  Manual. — The  last  twelve  pages  of  the  book  are 
devoted  to  notes,  remarks,  suggestions,  and  hints  to  teachers ;  and  to 
this  Manual  frequent  references  are  made  in  the  body  of  the  work. 
Teachers  should  not  fail  to  consult  the  Manual,  whenever  reference 
is  made  to  it. 

The  many  new  and  valuable  features  of  the  book,  its  superior 
typography  and  beautiful  illustrations,  its  great  number  of  practical 
problems  drawn  directly  from  business  life,  and  its  adaptation  to  the 
wants  of  graded  schools,  and  to  the  capacity  of  beginners  in  schools 
of  any  grade,  will,  it  is  hoped,  secure  for  it  the  attention  and  careful 
examination  of  intelligent,  progressive  teachers. 


iVBE 


CONTENTS. 


CHAPTER  I.— INTEGERS. 

PAGE 

Section        I.— Notation  and  Numeration 9 

Section      II. — ^Addition 17 

Section     III. — Subtraction 31 

Section     IV. — Multiplication 45 

Section       V. — Division 64 

Section     VI. — Measurements 83 

Section    VII. — Problems  in  Integers 89 

CHAPTER  II. -DECIMALS. 

Section        I. — ^Notation  and  Numeration 94 

Section      II. — Addition 101 

Section     III. — Subtraction 105 

Section     IV, — Multiplication 108 

Section       V. — Division 113 

Section     VI.— United  States  Money 121 

Section    VII. — Problems  in  Decimals 125 

CHAPTER  III. -COMPOUND  NUMBERS. 

Section        I. — Notation  and  Reduction 129 

Section      II. — Addition 149 

Section     III. — Subtraction 151 

Section     IV. — Multiplication 155 

Section       V. — Division 157 

Section      VI. — Problems  in  Compound  Numbers 159 


YIU 


CONTENTS 


CHAPTER  IV.-FRACTIONS. 

PAGB 

Section        I. — Induction  and  Notation 163 

Section       II.— Reduction 166 

Section     III. — Addition 171 

Section     IV.— Subtraction 174 

Section      V. — Multiplication 177 

Section     VI. — Division 181 

Section   VII. — Cancellation 184 

Section  VIII. — Problems  in  Fractions 186 

CHAPTER  V. -PERCENTAGE. 

Section        I. — Notation  and  Numeration 189 

Section       II. — General  Applications 190 

Section     III. — Commission 191 

Section     IV. — ^Insurance 192 

Section       V.— Profit  and  Loss 193 

Section     VI. — Interest 195 

Section    VII.— Problems  in  Percentage 199 

CHAPTER  VI. 
Miscellaneous  Problems 201 

Manual  for  Teachers 209 


top. 

ElEMENfm 'ARITHMETIC. 


CHAPTER  I. 
INTEGERS. 

SECTION  I.  * 

1.  In  writing  numbers,  ten  characters,  called  figures, 
are  used. 

The  first  figure,  0,  is  called  a  cipher  or  rmught^  and 
denotes  nothing  or  the  absence  of  number. 

The  other  nine  figures  are  used  to  represent  the 
first  nine  numbers. 


7.  Seven. 


9.     Nine. 


10  INTEGERS. 

To  express  numbers  greater  than  nine,  two  or  more 
of  these  ten  figures  must  be  combined. 

2.  In  writing  numbers,  every  ten  ones  taken  together 
are  called  a  ten. 

Ten  is  written  10 

Two  tens    are  called  twenty,  written,  20 


Three  tens 

a 

'  u 

thirty, 

SO 

Four  tens 

a 

u 

forty, 

Jfi 

Five  tens 

u 

ii 

fifty. 

50 

Six  tens 

u  ' 

u 

sixty, 

60 

Seven  tens 

u 

ii 

seventy. 

70 

Eight  tens 

u 

ii 

eighty, 

80 

Nine  tens 

u 

u 

ninety, 

90 

When  two  figures  are  written  together  to  express  a 
number,  the  left-hand  figure  expresses  tens,  and  the 
right-hand  figure  ones. 

Sixteen        consists  of  1  ten  and  6  ones,  written  16 


Twenty-four 

u 

"  3  tens 

u 

4    " 

a 

n 

Thirty-two 

a 

"  3     " 

u 

2     " 

a 

32 

Forty-nine 

u 

a   ^      u 

a 

9     " 

a 

JiS  . 

Fifty-five 

(( 

a    5      a 

u 

5     " 

(( 

55 

Sixty-seven 

u 

a    (5      u 

a 

7     " 

11 

67 

Seventy-three 

a 

u    ly      a 

u 

3     " 

a 

73 

Eighty 

a 

u    3      u 

u 

0     " 

a 

80 

Ninety-one 

u 

u    9      u 

ii 

1  one, 

(See 

Manual, 

91 
page  214.) 

JEXEMCISES. 

1.  Write  in  words  the  following  numbers :  14,  25,  37,  42, 
56,  69,  71,  88,  93. 

2.  Express  by  figures  the  following  numbers:  twelve, 
twenty-eight,  thirty-five,  forty-one,  fifty-nine,  sixty-three, 
seventy-six. 

3.  Write  in  words,  17,  29,  30,  48,  52,  65,  70,  81,  99. 

4.  Express  by  figures,  forty-three,  sixty-six,  ninety-five, 
fifteen,  eighty-six,  thirty-eight,  fifty-seven,  sixty-one. 

5.  Write  in  figures,  twenty,  forty,  eleven,  thirty-six,  ninety- 
four,  eighty-nine,  forty-six,  seventy-five. 


NOTATION     AND     NUMERATION.  11 

6.  Write  in  words,  13,  45,  51,  78,  83,  97. 

7.  Write  in  words,  60,  91,  79,  84,  27,  83. 

8.  Express   by  figures,   twenty-two,  thirty-nine,  fifty-four, 
ninety-six,  twenty-seven,  sixty-two,  fifty-three,  seventy-four. 

9.  Write  in  figures,    eighty-seven,  ninety-two,  thirty-four, 
twenty-six,  seventy-two,  sixty-eight,  forty-four,  ninety-eight. 

10.  Write  in  words,  19,  7)^171,  64,  23,  82,  58. 


3*  In  writing  numbers,  every  ten  tens  taken  together 
are  called  a  hundred. 

One  hundred  is  written  100 

Twenty  tens  are  two  hundred,  written  200 


Thirty  tens      " 

three  hundred,      ' 

'       SOO 

Forty  tens       " 

four  hundred,       ' 

'       400 

Fifty  tens        " 

five  hundred,        ' 

'       500 

Sixty  tens        " 

six  hundred,         ' 

'       600 

Seventy  tens    " 

seven  hundred,     ' 

'       700 

Eighty  tens     " 

eight  hundred,      ' 

'       800 

Ninety  tens     " 

nine  hundred,       ' 

'       900 

When  three  figures  are  written  together  to  express  a 
number,  the  left-hand  figure  expresses  hundreds,  the 
second  or  middle  figure,  tens,  and  the  right-hand  figure, 
ones.  Thus,  two  hundred  forty-three  consists  of  2  hun- 
dreds, 4  tens,  and  3  ones,  and  is  written  243. 

The  numbers  in  the  first  column  below  consist  of 
hundreds,  tens,  and  ones,  as  shown  in  the  second  col- 
umn, and  are  written  as  in  the  third  column. 

One  hundred  forty-nine,    1  hundred,   4  tens,  and  9  ones,    I49 


Four  hundred  sixty-two. 

4  hundreds 

,  6    "        "2 

"       462 

Five  hundred  twenty. 

5 

2    "        "0 

"       520 

Six  hundred  seventy. 

6 

7    "        "0 

"        670 

Seven  hundred  five, 

7 

0    "        "5 

"        705 

Eight  himdred  four. 

8 

0    "        "4 

"        8O4 

Nine  hundred  twelve. 

9 

Iten,      "    2 

(See  Manual, 

"        912 

page  214.) 

12  INTEGERS. 

exehcis  ES. 

11.  Write  in  words  the  numbers  247,  356,  528,  646,  935. 

12.  Express  by  figures,  one  hundred  seventy-three,  four 
hundred  ninety-one,  seven  hundred  sixty-four,  and  nine  hun- 
dred eighty-two. 

13.  Write  in  words,  617,  121,  745,  514,  311. 

14.  Express  by  figures,  four  hundred  nineteen,  nine  hundred 
thirty-nine,  three  hundred  thirty-three,  eight  hundred  eleven. 

15.  Write  in  words,  560,  310,  290,  420,  750. 

16.  Write  in  figures,  one  hundred  thirty,  six  hundred  forty, 
eight  hundred  eighty,  four  hundred  ten. 

17.  Write  in  words,  208,  906,  301,  606,  807. 

18.  Express  by  figures,  eight  hundred  two,  one  hundred 
nine,  four  hundred  three,  seven  hundred  five. 

19.  Write  in  words,  293,  780,  519,  103,  612,  999. 

20.  Express  by  figures  the  numbers  which  consist  of  8  hun- 
dreds, 2  tens,  and  4  ones ;  2  hundreds,  1  ten,  and  8  ones ;  5 
hundreds,  7  tens,  and  0  ones ;  9  hundreds,  5  tens,  and  4  ones ; 
3  hundreds,  0  tens,  and  7  ones. 

4*  In  writing  numbers,  every  ten  hundreds  taken  to- 
gether are  called  a  thousand.  Thousands  are  written 
thus  : 

One  thousand,      1000 

Two  thousand,    2000 

Three  thousand,  3000 

Four  thousand,   Ji,000 


Five  thousand,  5000 
Six  thousand,  6000 
Seven  thousand,  7000 
Eight  thousand,  8000 


Nine  thousand,  9000 

In  any  number  written  with  more  than  three  figures, 
the  figure  at  the  left  of  hundreds  expresses  thousands. 
Thus,  3579  consists  of  3  thousands,  5  hundreds,  7  tens, 
and  9  ones  ;  and  expresses  the  number  three  thousand 
five  hundred  seventy-nine. 

'   5*  Every  ten  thousands  taken  together  are  called  a 
ten-thousand. 

6*  Every  ten  ten-thousands  taken  together  are  called 
a  hundred-thousand. 


NOTATION     AND    NUMERATION.  13 

When  a  figure  stands  at  the  left  of  thousands,  it  ex- 
presses ten-thousands  ;  and  when  a  figure  stands  at  the 
left  of  ten-thousands,  it  expresses  hundred-thousands. 
Ten  thousand  is  written  10000 

Two  ten-thousands  are  twenty  thousand,  written  20000 
Three  ten-thousands  are  thirty  thousand,  "  30000 
Eight  ten-thousands  are  eighty  thousand,  "  80000 
One  hundred-thousand  is  written  100000 

Two  hundred  thousands  are  written  200000 

Five  hundred  thousands  "        "  500000 

7.  Every  three  figures  in  any  number  counting  from 
the  right  are  called  a  Period.     Periods 
of   figures    are    separated   from   each      I  | 
other  by  commas.  ^^La^^^ 

The  first  or  right-hand  period  con-        8  7  4,235 
sists  of  ones,  tens,  and  hundreds ;  and 
the  second  period  of  ones,  tens,  and 
hundreds  of  thousands.     Thus, 

Eighteen  thousand  five  hundred  thirty-six  is  written 

Thirty-two  thousand  eight  " 

Forty-seven  thousand  two  hundred  " 

Sixty  thousand  four  hundred  twenty  " 

Two  hundred  forty  thousand  " 

Four  hundred  eight  thousand  five  hundred  " 

Five  hundred  thousand  three  hundred  sev- 
enty-five " 

Six  hundred  fifty-two  thousand  ten  " 

Seven  hundred  forty-four  thousand  six        " 

Eight  hundred  fifty-three  thousand  five 
hundred  seventy-six  "        "       853^576 

(See  Manual,  page  214.) 
EXEMCISJES. 

21.  Write  in  words,  8,000;  5,400;  2,560;  1,644;  3,739. 

22.  Write  in  words,  6,944 ;  3,405 ;  4,094 ;  7,010 ;  6,009. 

23.  Express  by  figures,  five  thousand  three  hundred  fifty- 
six,  seven  thousand  two  hundred  forty,  one  thousand  nine 
hundred  three,  four  thousand  fifty-six. 


t  : 
1  s 

:     £    :   : 

i     1   3   1 

J  ^ 

§     5  5   § 

ivritt( 

3n     18,536 

u 

32,008 

u 

Jft,200 

u 

60,^20 

u 

21^0,000 

u 

1^08,500 

u 

500,375 

u 

652,010 

u 

7U,006 

14  INTEGERS. 

24.  Express  by  figures,  nine  thousand  eight  hundred,  two 
thousand  six,  eight  thousand  fifty,  five  thousand,  six  thousand 
eight  hundred  nineteen,  one  thousand  one. 

25.  Write  in  words,  15,380 ;  26,506  ;  37,081 ;  40,269 ;  98,274. 

26.  Write  in  words,  6,793;  72,400;  80,560  ;  63,004;  50,041. 

27.  Write-in  words,  33,000;  40,900;  90,209;  19,040;  20,007. 

28.  Express  by  figures,  sixty  thousand,  fifty  thousand 
twenty,  nineteen  thousand  three  hundred  one,  fifty-six  thou- 
sand eleven. 

29.  Write  in  words,  132,041 ;  270,405  ;  320,500  ;  400,385. 

30.  Write  in  words,  574,000 ;  629,005  ;  700,044 ;  803,000 ; 
957,503  ;  793,461 ;  809,051 ;  907,200. 

31.  Write  in  words,  503,020  ;  482,070  ;  600,002 ;  855,480. 

32.  Write  in  words,  100,905  ;  350,240  ;  904,306  ;  100,040. 

33.  Express  by  figures,  one  hundred  five  thousand  four  hun- 
dred seventy,  two  hundred  thousand,  five  hundred  forty  thou- 
sand seventy-two,  seven  hundred  forty-seven  thousand  two 
hundred. 

34.  Express  by  figures,  two  hundred  fifty  thousand  three 
hundred  sixty-three,  four  hundred  sixty  thousand  twenty,  seven 
hundred  ten  thousand,  eight  hundred  one  thousand  four. 

35.  Express  by  figures,  two  hundred  thousand  six  hundred 
forty,  three  hundred  five  thousand  two  hundred  ninety-four, 
six  hundred  eighty  thousand  five,  nine  hundred  thousand  six 
hundred. 


a  § 


8.  The  third  period  of  figures 
consists  of  ones,  tens,  and  hundreds 
of  millions.     Thus,  ^7^  ^  5~0~8 ,294 

In  any  full  period  the  right- 
hand  figure  is  ones,  the  middle 
figure  is  tens,  and  the  left-hand 
figure  is  hundreds.  Thus,  in  any  number  consisting 
of  three  full  periods,  there  are  ones,  ones  of  thousands, 
and  ones  of  millions  ;  tens,  tens  of  thousands,  and  tens 
of  millions  ;  and  hundreds,  hundreds  of  thousands,  and 
hundreds  of  millions. 


NOTATION     AND     NUMERATION, 


Two  million  five  hundred  thousand  eighty  is  written  2, 500, 080 

Thirty-four  million  three  hundred  twenty- 
four  thousand  five  hundred  eighty-six  "      "      34,324,586 

Forty  million  forty-four  thousand  twelve  "      "      40,044^012 

One  hundred  twenty-nine  million  three 
hundred  seventeen  thousand  five  hun- 
dred " 

Six  hundred  fifty  million  two  hundred 
thousand  seventy  " 

Nine  hundred  three  million  fifty  thousand 
five  hundred  ninety-four,  " 

Three  hundred  million  three  thousand 
thirty  " 


129,317,500 
650,200,070 
903,050,594 

300,003,030 

9.  The  writing  of  mimbers  in  figures  is  Notation. 

10.  The  reading  of  numbers  which  are  expressed  by 
figures  is  Numeration. 

11,  The  place  which  any  figure  occupies  in  a  number 
determines  the  value  expressed  by  it  in  that  number. 

The  values  of  the  different  places  in  their  order  are 
shown  by  the  following 

RATION     TABLE. 

ten, 

hundred, 

thousand, 

ten-thousand, 

hundred-thousand, 

million, 

ten-million, 

hundred-million. 

ones, 

tens, 

hundreds, 

thousands, 

ten-thousands, 

hundred-thousands, 

millions, 

ten-millions. 

(See  Manual,  page  214.) 


NOTATION     AND     N 

10  ones 

10  tens 

10  hundreds 

10  thousands 

10  ten-thousands 

10  hundred-thousands 

10  millions 

10  ten-millions 

1  ten 

1  hundred 

1  thousand 

1  ten-thousand 

1  hundred-thousand 

1  million 

1  ten-million 

1  hundred-million 


UME 

are  1 

"  1 

"  1 

"  1 

"  1 

"  1 

"  1 

"  1 


is  10 
"  10 
"  10 
"  10 
"  10 
"  10 
"  10 
"  10 


16  INTEGEES. 


MEVIEW    EXERCISES. 

36.  Write  in  words,  4,650 ;  738 ;  450,840 ;  93,066 ;  8,050,800; 
1,005;  4,000,800. 

37.  Write  in  words,  37,098,430 ;  502,000 ;  730,900. 

38.  Write  in  words,  85,700,035  ;  6,000,030  ;  13,006,400 ; 
45,000,000;  6,413;  2,578,034. 

39.  Write  in  words,  73,059,209 ;  10,765,291 ;  8,010;  70,045; 
8,050,000. 

40.  Express  by  figures,  seven  hundred  six  thousand  two 
hundred  ninety-one,  seventy-four  thousand  seven  hundred 
four,  nine  thousand  ninety,  eighty  thousand  twelve. 

41.  Express  by  figures,  eight  million  five  thousand  three 
hundred  ninety-four,  six  million  eight,  seven  million,  three 
hundred  thousand  seven  hundred  twenty. 

43.  Express  by  figures,  nine  million  two  hundred  seventy, 
two  hundred  three  thousand  four  hundred  five,  twenty  thou- 
sand six  hundred  seven,  eighteen  million  nineteen. 

43.  Write  in  words,  4,080,306;  332,107,003;  14,200;  500,007; 
60,572;  536,000. 

44.  Write  in  words,  34,000,709;  1,702,050;  605,400,300; 
9,600,309. 

45.  Write  in  words,  12,065,587  ;  40,080,276  ;  7,200,000  ; 
15,009,820;  3,031,504. 

46.  Express  by  figures,  four  million  two  hundred  fifty-nine, 
seven  million  two  hundred  two  thousand  five  hundred,  eighty 
thousand,  four  hundred  thousand  two  hundred  fifty-six. 

47.  Express  by  figures,  eight  hundred  million  seven  hun- 
dred seven  thousand  five  hundred  six,  sixteen  thousand  six- 
teen, seven  million  five  thousand  forty-four,  twenty-nine 
million  forty-one  thousand. 

48.  Express  by  figures,  two  hundred  four  thousand  two 
hundred  seventy,  fifty  thousand  thirty-three,  one  million  three 
hundred  six,  seven  hundred  fifty  thousand  nine. 

49.  Write  in  figures,  ten  million  twenty-five  thousand  four 
hundred,  six  thousand  one,  three  million  two  thousand,  six- 
teen million  ninety  thousand  five. 

50.  Write  in  words,  7,019,003 ;  506,427,  711 ;  243,424 ; 
736,378;  9,999;  300,003,303. 


ADDITION. 


17 


SECTION  II. 
i:isri>TJCTio:Nr. 

(See  Manual,  page  215.) 

12.  Heee  is  a  picture  of  some  boys  and  girls  in  an 
orcliard  gathering  apples.  John  has  1  apple  in  each 
hand ;  Harry  has  2  apples  in  one  hand  and  1  in  the 
other  ;  Mary  has  3  apples  in  one  hand  and  1  in  the  other; 
and  Fanny  has  4  apples  in  her  lap  and  1  in  her  hand. 

1.  How  many  apples  has  John  ?  How  many  has  Harry  ? 
How  many  has  Mary  ?     How  many  has  Fanny  ? 

2.  How  many  apples  have  John  and  Harry  together  ? 

3.  How  many  apples  have  Mary  and  Fanny  ? 

4.  If  John  and  Harry  give  their  apples  to  Mary,  how  many 
apples  will  she  have  ? 

5.  If  Harry  and  Mary  give  their  apples  to  Fanny,  how  many 
will  she  have  ? 


18  INTEGERS. 

6.  If  all  the  children  put  their  apples  into  Mary's  basket, 
how  many  apples  will  be  in  the  basket  ? 

13.  When  two  or  more  numbers  are  united  to  form 
one  number,  the  process  is  Addition.  * 

14.  The  result  thus  formed  is  the  Amount  or  Sum,  and 
the  numbers  to  be  united  are  the  Parts. 

The  amount  or  sum  must  contain  as  many  ones  as 
all  the  parts  taken  together. 

7.  What  is  the  amount  of  11  cents,  5  cents,  and  8  cents  ? 

8.  What  is  the  amount  of  8  pencils,  6  pencils,  and  10  pencils  ? 

9.  What  is  the  sum  of  4  walnuts,  1  walnut,  2  walnuts,  and 
9  walnuts  ? 

10  What  is  the  sum  of  2  days,  4  days,  3  days,  and  6  days  ? 

11.  Add  8  peaches,  4  peaches,  3  peaches,  and  7  peaches. 

12.  Add  5  roses,  12  roses,  7  roses,  and  6  roses. 

13.  Martha  has  a  5-cent  piece,  a  3-cent  piece,  a  2-cent  piece, 
and  a  10-cent  piece.     What  sum  of  money  has  she  ? 

14.  A  laborer  worked  four  weeks,  earning  8  dollars  the  first 
week,  6  dollars  the  second,  4  dollars  the  third,  and  7  dollars 
the  fourth.     What  amount  of  money  did  he  earn 

15.  Add  3  and  7  and  5  and  8  and  2. 

16.  Add  9  and  4  and  6  and  1  and  5. 

15.  This  sign  +,  written  between  numbers,  signifies 
that  they  are  to  be  added. 

It  is  called  Ptus,  or  the  Sign  of  Addition.  . 

16.  This  sign  =  written  between  numbers  or  sets  of 
numbers,  signifies  that  they  are  equal  to  each  other. 

It  is  called  the  Sign  of  Equality.     Thus, 

6  +  7  +  12  =  25  is  read,  6  plus  7  plus  12  equal  25. 

17.  Bead  7  +  3  +  5  =  15. 

18.  Bead  29  =  10  +  9  +  6  +  4. 

19.  Read  12  +  5  +  4  =  13  +  8. 

20.  16  eggs  +  5  eggs  +  11  eggs  +  9  eggs  =  how  many  eggs  ? 

21.  8  hats  +  21  hats  +  6  hats  +  10  hats  =  how  many  hats? 


ADDITION. 


19 


22.  What  is  the  sum  of  7  chairs  +  12  chairs  +  6  chairs  +  10 
cliairs  +  1  chair  ? 

23.  What  is  the  amount  of  19  dollars  +  10  dollars  +  7  dol- 
lars +  6  dollars  +  5  dollars  ? 

24.  18  +  7  +  6  +  9  =  how  many  ?      (See  Manual,  page  215.) 

17»     ADDITION     TABLE. 


rtJO    123456789 
W|o    000000000 

KJ0123456789 
0|5    555555555 

0123456789 

5    6    7    8    9  10  11  12  13  14 

-(0    123456789 

1)1    111111111 

123456789  10 

cJO    123456789 

0]6    666666666 

6    7    8    9  10  11  12  13  14  15 

oJO    123456789 
*]2    222222222 

^012345678    9 

7|7    777777777 

23456789  10  11 

7    8    9  10  11  12  13  14  15  16 

oJO    123456789 
33333333333 

q(0    123456789 
o]8    888888888 

3    4    5    6    7    8    9  10  11  12 

8    9  10  11  12  13  14  15  16  17 

-(0123456789 
4|4    444444444 

q(0    123456789 
»]9    999999999 

4    5    6    7    8    9  10  11  12  13 

9  10  11  12  13  14  15  16  17  18 

OMAJL    EXEMCISES, 

1. — 1.  Count  to  100  in  this  manner ;  0  and  1  are  1,  1  and  1  are  2, 

2  and  1  are  3,  and  so  on. 

2.  Count  to  100,  thus  ;  0, 1,  2,  3,  4,  5,  and  so  on. 

2. — ^1.  Count  by  2's  to  100,  in  this  manner,  0  and  2  are  2,  2  and  2 
are  4,  4  and  2  are  6,  and  so  on. 

2.  Count  by  2's  to  100,  thus ;  0,  2,  4,  6,  8,  and  so  on. 

3.  Commence  with  1,  and  count  by  2's  to  101,  thus ;  1  and  2  are  3, 

3  and  2  are  5,  5  and  2  are  7,  and  so  on. 

4.  Count  by  2's  from  1  to  101,  thus ;  1,  3,  5,  7,  9,  and  so  on. 

3. — 1.  Count  by  3's  from  0  to  102,  thus ;  0  and  3  are  3,  3  and  3  are  6, 
6  and  3  are  9,  and  so  on. 

2.  Count  by  3's  from  0  to  102,  thus  ;  0,  3,  6,  9,  12,  and  so  on. 

3.  Commence  with  1  and  count  by  3's  to  lOO,  thus  ;  1  and  3  are  4, 

4  and  3  are  7,  7  and  3  are  10,  and  so  on. 

4.  Count  by  3's  from  1  to  100,  thus ;  1,  4,  7, 10,  i3,  and  so  on. 

5.  Commence  with  2  and  count  by  3's  to  101,  thus ;  2  and  3  are  5, 

5  and  3  are  8,  8  and  3  are  11,  and  so  on. 

6.  Count  by  3's  from  2  to  101,  thus ;  2,  5,  8, 11, 14,  and  so  on. 


20  INTEGERS. 

4. — 1.  Commence  with  0  and  count  by  4's  to  100,  thus ;  0  and  4 
are  4,  4  and  4  are  8,  8  and  4  are  12,  and  so  on. 

2.  Count  by  4's  from  0  to  100,  thus ;  0,  4,  8, 12, 16,  and  so  on. 

3.  Commence  with  1  and  count  by  4's  to  101,  thus ;  1  and  4  are  5, 

5  and  4  are  9,  9  and  4  are  13,  and  so  on. 

4.  Count  by  4's  from  1  to  101,  thus ;  1,  5,  9, 13, 17,  and  so  on. 

5.  Commence  with  2  and  count  by  4's  to  102,  thus ;  2  and  4  are  G, 

6  and  4  are  10, 10  and  4  are  14,  and  so  on. 

6.  Count  by  4's  from  2  to  102,  thus ;  2,  6, 10, 14, 18,  and  so  on. 

7.  Commence  with  3  and  count  by  4's  to  103,  thus ;  3  and  4  are  7, 

7  and  4  are  11, 11  and  4  are  15,  and  so  on. 

8.  Count  by  4's  from  3  to  103,  thus ;  3,  7, 11,  15, 19,  and  so  on. 

5. — 1.  Commence  with  0  and  count  by  5's  to  100,  thus ;  0  and  5 
are  5,  5  and  5  are  10, 10  and  5  are  15,  and  so  on. 

2.  Count  by  5's  from  0  to  100,  thus ;  0,  5,  10, 15,  and  so  on. 

3.  Commence  with  1  and  count  by  5's  to  101.    (See  Manual,  page  215.) 

4.  Commence  with  2  and  count  by  5's  to  102. 

5.  Count  by  5's  from  3  to  103. 

6.  Count  by  5's  from  4  to  104. 

6. — 1.  Count  by  6's  from  0  to  102,  thus ;  0  and  6  are  6,  6  and  6  are 
12, 12  and  6  are  18,  and  so  on. 

2.  Count  by  6's  from  0  to  102,  thus ;  0,  6, 12, 18,  24,  and  so  on. 

3.  Commence  with  1  and  count  by  6's  to  103. 

4.  Count  by  6's  from  2  to  104. 

5.  Count  by  6's  from  3  to  105. 

6.  Count  by  6's  from  4  to  100. 

7.  Count  by  6's  from  5  to  101.  , 

7, — 1.  Commence  with  0  and  count  by  7'8  to  105,  thus ;  0  and  7 
are  7,  7  and  7  are  14, 14  and  7  are  21,  and  so  on. 

2.  Count  by  7's  from  0  to  105,  thus ;  0,  7, 14,  21,  28,  and  so  on. 

3.  Commence  with  1  and  count  by  7's  to  106. 

4.  Commence  with  2  and  count  by  7's  to  100. 

5.  Commence  with  3  and  count  by  7*8  to  101. 

6.  Count  by  7's  from  4  to  102. 

7.  Count  by  7's  from  5  to  103. 

8.  Count  by  7's  from  6  to  104. 

8. — 1.  Commencing  with  0,  count  by  8's  to  104,  thus ;   0  and  8 
are  8,  8  and  8  are  16, 16  and  8  are  24,  and  so  on. 

2.  Count  by  8's  from  0  to  104,  thus ;  0,  8, 16,  24,  32,  and  so  on. 

3.  Commencing  with  1,  count  by  8's  to  105. 

4.  Commencing  with  2,  count  by  8's  to  106. 

5.  Commencing  with  3,  count  by  8's  to  107. 

6.  Count  by  8's  from  4  to  100. 

7.  Count  by  8'8  from  5  to  101. 

8.  Count  by  8'8  from  6  to  102. 

9.  Count  by  S's  from  7  to  103. 


ADDITION.  21 

9. — 1.  Commencing  with  0,  count  by  9's  to  108,  thus ;  0  and  9  are  9, 
9  and  9  are  18, 18  and  9  are  27,  and  so  on. 

2.  Count  by  9's  from  0  to  108,  thus ;  0,  9, 18,  27,  36,  and  so  on. 

3.  Commencing  with  1,  count  by  9's  to  100. 

4.  Commencing  with  2,  count  by  9's  to  101. 

5.  Commencing  with  3,  count  by  9's  to  102. 

6.  Count  by  9's  from  4  to  103. 

7.  Count  by  9's  from  5  to  104. 

8.  Count  by  9's  from  6  to  105. 

9.  Count  by  9's  from  7  to  106. 

10.  Count  by  9's  from  8  to  107. 


C^SE     I. 
The  aum  of  all  the  figures  of  any  place  not  more  than  9. 

18.  ExiiMPLE.  What  is  the  sum  of  2,344  and  3,152  ? 

Explanation. — Since  these  parts  are  too  solution. 
large  to  be  added  mentally,  we  write  them     2,344 1  ^^^^^ 
one  under  the  other,  writing  the  ones  of     ^>1^^  ; 
one  part  under  the  ones  of  the  other,  the     5,496    Sum. 
fens  under  tens,  the  hundreds  under  hun- 
dreds, and  the  thousands  under  thousands.     The  sum  of 
2  ones  »nd  4  ones  is  6  ones,  which  we  write  under  the 
ones;  the  sum  of  5  tens  and  4  tens  is  9  /ens,. which  we 
write  under  the  tens  ;  the  sum  of  1  hundred  and  3  hun- 
dreds is  4  hundreds,  which  we  write  under  the  hundreds; 
and  the  sum  of  3  thousands  and  2  thousands  is  5  thou- 
sands, which  we  write  under  the  thousands.     The  result, 
5,496,  is  the  sum  required. 

19.  We  can  add  apples  to  apples,  dollars  to  dollars, 
pens  to  pens,  or  hours  to  hours ;  but  we  can  not  add 
apples  to  dollars,  nor  pens  to  hours.  For  4  apples  +  9 
dollars  —  neither  13  apples  nor  13  dollars. 

Again,  we  can  add  ones  to  ones,  tens  to  tens,  or  hun- 
dreds to  hundreds  ;  but  we  can  not  add  ones  to  hun- 
dreds, nor  tens  to  thousands.  For  4  tens  +  9  thou- 
sands =  neither  13  tens  nor  13  thousands.     Hence, 


22 


INTEGEES 


20»   General  Principles  o/ Addition, 

I.  Only  numbers  expressing  the  same  hind  of  things  can 
he  added. 

II.  Only  figures  occupying  the  same  place  in  different 
numbers  can  be  added  ;  that  is,  ones  must  be  added  to 
ones,  tens  to  tens,  hundreds  to  hundreds,  thousands  to 

thousands,  and  so  on.       (See  Manual,  page  215.) 
PROBIjEMS. 

Find  the  sum  of  the  numbers  in  each  of  the  first  ten  problems. 
(1)  (2)  (3)  (4)  (5)  (6) 

62  26  34  452  281  504 

24  72  145  37  612  283 


C^) 

(8) 

(9) 

(10) 

235  men 
612  men 
141  men 

2,413  books 

146  books 

30  books 

5,241  miles 

306  miles 

2,432  miles 

31,410  dollars 

1,245  dollars 

26,332  dollars 

11.  James  paid  12  cents  for  a  slate,  and  15  cents  for  a  writ- 
ing-book.    How  many  cents  did  he  pay  for  both  ? 

12.  Myron  found  25  plums  under  one  tree  in  the  garden, 
and  13  ptums  under  another.  How  many  plums  did  he  find 
under  both  trees  ? 

13.  In  a  village  school  are  56  boys  and  43  girls.  How  many 
pupils  in  the  school  ? 

14.  One  day  a  lady  traveled  42  miles  by  railroad  and  16 
miles  by  stage.     How  many  miles  did  she  travel  ? 

15.  An  orchard  consists  of  53  winter  apple-trees  and  14  fall 
apple-trees.     How  many  trees  are  in  the  orchard  ? 

16.  A  builder  paid  610  dollars  for  a  city  lot,  and  built  upon 
it  a  house  which  cost  him  2,085  dollars.  How  much  did  the 
house  and  lot  cost  ? 

17.  What  is  the  sum  of  542  +  36  ? 

18.  21  -f-  45  -f  32  =  how  many? 

19.  6,132  +  31  +  36  =  how  many?  6,199. 


ADDITION.  23 

20.  What  is  the  sum  of  123  +  231  +  312  + 123  +  201  ?    989. 

21.  What  is  the  sum  of  four  hundred  one  thousand  nine 
hundred  fifty,  twenty-four  thousand  twenty-four,  and  two 
thousand  and  four  ?  4^7, 978. 

22.  A  farmer  harvests  from  five  fields  of  wheat,  151  bushels, 
204  bushels,  120  bushels,  312  bushels,  and  211  bushels.  How^ 
many  bushels  of  wheat  did  he  harvest  ?  998. 

23.  In  January  a  laborer  deposited  in  the  savings  bank  12 
dollars,  in  February  30  dollars,  in  March  13  dollars,  in  April 
11  dollars,  in  May  21  dollars,  and  in  June  12  dollars.  How 
many  dollars  did  he  deposit  in  the  six  months  ?    99  dollwrs. 

24.  One  week  in  May  one  dairyman  furnished  to  a  cheese 
factory  2,432  pouncte  of  milk,  another  dairyman  4,145  pounds, 
and  another  3,221  pounds.  How  many  pounds  were  furnished 
by  the  three  dairymen  ?  9, 798. 

25.  The  amount  of  cheese  manufactured  at  the  same  factory 
in  June  was  12,147  pounds,  in  July  13,410  pounds,  in  August 
22,221  pounds,  and  in  September  11,211  pounds.  How  many 
pounds  were  manufactured  in  the  four  months  ?        58,989. 

26.  At  a  cotton  factory  1,465,207  yards  of  cloth  were  made 
in  1864,  and  1,532,492  yards  in  1865.  How  many  yards  were 
made  in  the  two  years  ?  2,997,699. 

27.  A  grocer  bought  four  hogsheads  of  sugar,  weighing 
1,154  pounds,  1,213  pounds,  1,301  pounds,  and  1,231  pounds. 
How  many  pounds  did  they  all  weigh  ?  4,899. 

28.  Long  Island  consists  of  three  counties.  Kings  County 
contains  72  square  miles.  Queens  County  410  square  miles, 
and  Suffolk  County  1,200  square  miles.  How  many  square 
miles  in  the  island  ?  i^  682. 

29.  A  fruit-grower  sold  123  barrels  of  apples  to  one  man, 
31  barrels  to  another,  103  to  a  third,  30  barrels  to  a  fourth, 
and  112  barrels  to  a  fifth.  How  many  barrels  of  apples  did 
he  sell  ?  S99^ 

30.  The  mason  work  of  a  new  school-house  cost  1,220  dol- 
lars, the  carpenter  work  1018  dollars,  and  the  painting  and 
glazing  430  dollars.  How  many  dollars  did  the  school-house 
cost?  ^2,668. 


24  INTEGERS. 

CA.SE3    II. 
The  sum  of  all  the  figures  of  any  place  more  than  9. 

FIRST     METHOD. 

21.  Ex.  What  is  the  sum  of  28,  76  and  39  .^ 

Explanation.  —  IsL   Writing  the  numbers.  —  first  step. 
AVe  write  ones  under  ones,  and  tens  under      ^^ 
tens,  and  below  the  last  number  we  draw  two       '  ^ 
parallel  horizontal  lines,  far  enough  apart  to      — 
allow  us  to  write  figures  between  them.  — 

2d.  Adding  the  Numbers. — Adding  i;he  ones,  we  find 
the  sum  to  be  23,  or  3  ones  and  2  tens.  We  second  step. 
write  the  3  ones  below  the  lower  line  as  the  28 
ones  of  the  required  sum,  and,  since  we  must  76 
add  the  2  tens  to  the  tens  of  the  given  num-  ?5 
bers,  we  write  them  in  tens'  place,  between  the  ?_ 
two  hues.     Adding  the  tens,  we  find  the  sum  ^ 

to  be  14,  or  4  tens  and  1  hundred.     As  there  are  no 
hundreds  in  the  given  numbers  to  which     solution. 
to  add  this  1  hundred,  we  write  the  4         28) 
tens  and  the  1  hundred  below  the  lower        76  [-  Parts. 
line,  as  tens  and  hundreds  of  the  required       _39 ) 
sum.      The  result,  143,  is  the   sum  re-        2 

quired.      (See  Manual,  page  215.)  143 

FMOBJOJEMS. 

31.  A  farmer  has  46  sheep  in  one  flock  and  38  in  another. 
How  many  sheep  has  he  ?  84. 

32.  A  merchant  sold  13  yards  of  calico  to  one  lady,  14  yards 
to  another,  and  16  yards  to  another.  How  many  yards  did  he 
sell  to  the  three  ladies  ?  4^. 

33.  Two  wood-choppers  worked  together  through  the  win- 
ter. One  of  them  chopped  174  cords  of  wood  and  the  other 
167  cords.     How  many  cords  did  both  of  them  chop  ?    Sj^I, 


ADDITION. 


25 


<&&,)£:ji 


—MEADOW^ 


-..:ri-.-^...../i;  :i:  :|o;|,  ;  3d!--c^  r^etcfl 


r,§(S  <3»  d*  *  as*"'"" 

rrS  pRCHARO.o*'""  b^ 


34.  On  this  map  of  a  farm, 
how  many  acres  of  wood- 
land on  both  sides  of  Wil- 
low Pond  ?  (See  Manual  p.  215.) 

35.  How  many  acres  of 
pasture  on  both  sides  of 
Stony  Brook  ? 

36.  How  many  acres  are 
in  the  two  meadows  ? 

37.  How  many  acres  of 
tilled  land  does  the  farm 
contain  ?  71. 

How  many  acres  are  on  the  east 
side  of  Willow  Pond  and  Stony  Brook  ? 

39.  How  many  acres  on  the  west  side  ? 

40.  How  many  acres  in  the  farm  in- 
cluding Willow  Pond  ?  256. 

41.  The  orchard  is  47  rods  long  and  34  rods  wide.  How 
many  rods  long  is  the  stone  fence  around  it  ?  162. 

42.  The  meadow  north  of  the  orchard  is  66  rods  long  and 
58  rods  wide.     How  many  rods  of  fence  around  it  ? 

43.  The  whole  pasture  is  110  rods  long  and  64  rods  wide. 
How  many  rods  of  rail  fence  on  the  three  sides,  as  shown  on 
the  map  ?  28^. 

44.  The  yard  and  garden  are  34  rods  long  and  19  rods  wide. 
In  front  is  a  picket  fence,  and  on  the  other  three  sides  is  stone 
fence.     How  many  rods  long  is  the  stone  fence  ? 

45.  How  many  rods  in  the  fences  which  inclose  the  yard 
and  garden  ? 

46.  The  lengths  of  the  different  fences  inclosing  the  farm 
are  shown  on  the  map.  How  many  rods  of  these  are  stone 
fence  ?  237. 

47.  How  many  rods  are  rail  fence  ? 

48.  How  many  rods  of  fence  of  all  kinds  around  the  farm  ? 

49.  How  many  rods  of  road  in  front  of  this  farm  ? 

C 


Principal            Distances  ] 

Stations. 

in  miles. 

Boston,   .     . 

. 

Worcester,  . 

.    44 

Springfield, 

.     54 

Pittsfield,    . 

.     53 

Albany,  .     . 

.    49 

Albany,  .     . 

. 

Schenectady, 

.     17 

Utica,      .     . 

.     78 

Syracuse,     . 

.     52 

Rochester,   . 

.     82 

Batavia,  .     . 

.     32 

Buffalo,  .     . 

.     36 

INTEGERS 

Distances  hetween  Boston  and  St.  Louis. 

Principal            Distances 
Stations.              in  miles. 

Buffalo,  .     .     . 

Principal  Distances 
Stations.              in  miles. 

Chicago, .     .    . 

Dunkirk,     .     . 

40 

Joliet,     ...     36 

Erie,  .... 

48 

Bloomington,  .     88 

Cleveland,  .     . 
Sandusky  City, 
Toledo,  .     .     . 

95 
61 
51 

Springfield,.  .  60 
Alton,  ...  72 
St.  Louis,     .     .    25 

Toledo,  .     .     . 
Adrian,  .     .     . 

32 

(See  Manual,  page  215.) 

Coldwater,  .     . 

56 

South  Bend,     . 

69 

La  Porte,     .     . 

27 

Chicago, .    .     . 

59 

50.  How  many  miles  from  Boston  to  Albany  ?  SOO. 

51.  How  many  miles  from  Albany  to  Buffalo  ?  297. 

52.  What  is  the  distance  from  Buffalo  to  Toledo  ? 

53.  What  is  the  distance  frem  Toledo  to  Chicago  ? 

54.  How  far  is  it  from  Chicago  to  St.  Louis  ?      281  miles. 

55.  How  far  is  it  from  Albany  to  Chicago  ?         835  miles. 

56.  What  is  the  distance  from  Boston  to  St.  Louis  ? 

57.  One  day  a  miller  bought  1,284  bushels  of  wheat,  and 
the  next  day  859  bushels.  How  many  bushels  did  he  buy  in 
the  two  days?  2,143. 

58.  A  butcher  killed  an  ox,  the  quarters  of  which  weighed 
respectively  136  pounds,  143  pounds,  178  pounds,  and  187 
pounds.     What  was  the  weight  of  the  four  quarters  ?     644- 

59.  A  grocer  bought  five  jars  of  butter,  containing  respec- 
tively 33  pounds,  47  pounds,  32  pounds,  54  pounds,  and  45 
pounds.    How  many  pounds  of  butter  did  the  five  jars  contain  ? 

60.  One  month  a  woolen  manufacturer  paid  out  31,587  dol- 
lars for  stock,  and  23,476  dollars  for  labor.  How  many  dol- 
lars did  he  pay  out  during  the  month  ?  55,063. 

61.  53  feet  +  171  feet  +  23,869  feet  +  24  feet  +  359,487 
feet  =  how  many  feet  ?  383,604  feet. 


ADDITION 


27 


SOLtTTlON. 


SECOND     METHOD. 

22.  Ex.  Add  346,  5,279,  and  8,165. 

Explanation. — After  writing  the  parts,  witli 
ones  under  ones,  tens  under  tens,  and  so  on,  ^^^ 

we  draw  one  horizontal  line  under  the  last        qSak 

number.     Adding  the  ones,  we  find  the  sum  to    '■ 

be  20,  or  0  ones  and  2  tens.  We  write  the  0  1^,790 
ones  below  the  line  in  the  ones'  place  of  the  required 
sum  ;  and  the  2  tens  we  add  with  the  tens  of  the  given 
numbers,  but  without  first  writing  it  in  a  line  by  itself. 
The  sum  of  all  the  tens  is  19,  or  9  tens  and  1  hundred. 
We  write  the  9  tens  below  the  line  as  the  tens  of  the 
required  sum  ;  and  the  1  hundred  we  add  with  the 
hundreds  of  the  given  numbers.  The  sum  of  all  the 
hundreds  is  7,  which  we  write  below  the  hne  in  hun- 
dreds' place.  The  sum  of  all  the  thousands  is  13, 
which  we  write  below  the  hne  as  the  thousands  and 
ten-thousands  of  the  required  sum.    The  result,  13,790, 

is  the  sum  required.      (See  Manual,  page  216.) 
:PItOBLEMS. 

63.  In  the  first  passenger  car  of  a  railroad  train  were  49 
passengers,  in  the  second  63,  in  the  third  54,  in  the  fourth  63, 
and  in  the  fifth  48.     How  many  passengers  were  on  the  train  ? 

63.  A  railroad  company  purchased  in  one  day  167  cords  of 
wood  at  one  station,  289  cords  at  another,  84  cords  at  another, 
and  417  cords  at  another.  How  many  cords  were  purchased 
at  the  four  stations  ?  957  cords. 

64.  A  merchant  by  selling  a  lot  of  damaged  goods  for  $587, 
lost  $94.     How  much  did  the  goods  cost  him  ?  $681. 

A  number  with  the  sign  $  before  it  expresses  dollars. 

65.  Three  men  engaged  in  business  together,  the  first  fur- 
nishing $3,425  dollars,  the  second  $2,163  dollars,  and  the  third 
$896.     What  was  the  amount  of  their  capital  ?  $6^Jf^4. 


28  INTEGERS. 

66.  A  grain-buyer  in  Chicago  paid  $7,594  for  a  cargo  of 
wheat,  shipped  it  to  New  York  at  an  expense  of  $2,841,  and 
sold  it  so  as  to  gain  $1,565.     For  how  much  did  he  sell  it  ? 

67.  A  merchant  pays  for  rent  of  store  1,275  dollars  a  year, 
for  clerk-hire  3,895  dollars,  for  fuel  242  dollars,  for  gas  437 
dollars,  for  freight  and  cartage  on  goods  936  dollars,  and  for 
other  expenses  359  dollars.  What  is  the  amount  of  his  yearly 
expenses  ?  $7, 144- 

68.  At  a  sale  of  government  vessels,  August  10,  1865,  the 
bark  Restless  was  sold  for  $12,000,  the  tug  Larkspur  for 
$8,100,  the  side- wheel  steamer  Alabama  for  $28,000,  the 
schooner  Matthew  Vassar  for  $7,300,  and  the  steam  packet- 
boat  Hartford  for  $9,100.  For  how  much  were  all  these  ves- 
sels sold?  $64,500. 

69.  A  store  in  a  brick  building  rents  for  $365  a  year,  the 
offices  in  the  second  story  rent  for  $162,  and  a  daguerrean 
room  in  the  third  story  rents  for  $78.  How  much  is  the  whole 
rent  of  the  building  ?  $605. 

70.  A  merchant's  cash  sales  on  Monday  were  $96,  Tuesday 
$132,  Wednesday  $98,  Thursday  $72,  Friday  $115,  and  Satur- 
day $149.  What  was  the  amount  of  his  cash  sales  for  the 
week  ?  $662. 

71.  One  season  a  farmer  killed  six  hogs  which  weighed  427 
pounds,  329  pounds,  314  pounds,  217  pounds,  208  pounds, 
and  1 96  pounds.  How  much  did  they  all  weigh  ?    1,691  pounds. 

72.  Seven  rafts  of  saw-logs  from  Alleghany  River  passed 
Pittsburg  in  one  day.  The  first  raft  contained  276  logs,  the 
second  359,  the  third  409,  the  fourth  293,  the  fifth  318,  the 
sixth  325,  and  the  seventh  358.  How  many  logs  in  all  the 
rafts?  2,338  has. 


(73) 

(74) 

(75) 

(76) 

(77) 

30,076 

141 

28 

14,193 

647,129,341 

5,821 

30,648 

52 

6,009 

327,293 

498 

8,291 

164 

417 

284,384 

167 

287 

386 

1,306 

43,100,085 

22,849 

165 

1,227 

129 

2,873 

3,482 

24 

2,873 

873 

541 

691 

2,841 

642 

154,685 

30,698 

482 

596 

578 

7,676 

28,165 

1,642 

417 

249 

48 

475 

56 

13,509 

3,871 

509 

1,465,127 

ADDITION.  29 

23*  ^ule  for  Addition  of  Integers, 

I.  Write  the  nuimhers  to  he  added  with  ones  under  ones, 
tens  under  tenSy  hundreds  under  hundreds,  and  so  on. 

n.  Add  the  column  of  ones,  and,  if  the  sum  does  not 
exceed  9,  place  it  under  the  ones ;  but  if  it  exceeds  9, 
place  the  right-hand  figure  under  the  ones. 

m.  Add  the  column  of  tens,  and  with  it  the  left-hand 
figure  of  the  sum  of  the  ones,  and  if  the  sum  does  not 
exceed  9,  place  it  under  the  tens  ;  hut  if  it  exceeds  9,  place 
the  right-hand  figure  under  the  tens. 

rV.  Proceed  in  the  same  manrwr  with  each  column  suc- 
cessively, and  write  down  the  whole  sum  of  the  left-hand 
column. 

PJROBLEMS. 

78.  A  builder  received  $17,525  for  erecting  a  church,  $2,485 
for  building  a  dwelling ;  $580  for  building  a  bam,  and  $265 
for  repairs  on  a  store.  How  much  did  he  receive  for  the  four 
jobs?  $20,855. 

79.  England  contains  57,101  square  miles,  Scotland  31,324 
square  miles,  Ireland  32,512  square  miles,  Wales  7,219  square 
miles,  and  the  smaller  British  islands  contain  324  square  miles. 
How  many  square  miles  in  the  whole  of  Great  Britain  ? 

80.  What  is  the  sum  of  thirty-five  million  eight  hundred 
seventy-six  thousand  one  hundred  twenty,  three  hundred 
ninety-six  thousand  four  hundred  ninety-one,  and  five  hundred 
forty-three  thousand  six  hundred  seven  ?  36,816,218. 

81.  One  year  a  farmer  raised  587  bushels  of  wheat,  1,229 
bushels  of  oats,  643  bushels  of  com,  184  bushels  of  rye,  259 
bushels  of  barley,  and  296  bushels  of  buckwheat.  How  many 
bushels  of  grain  did  he  raise  ?  3, 198  Imshels. 

83.  A  man  paid  $3,478  for  a  farm,  $1,117  for  live  stock, 
$635  for  farming  implements,  $423  for  grain  and  seeds,  and 
$189  for  repairing  fences  and  buildings.  How  much  was  his 
total  outlay  ?  $5,8Jt2, 


30  INTEGERS. 

83.  A  pork-packer  in  Cincinnati  packed  15,287  barrels  of 
pork  in  December,  13,164  barrels  in  January,  and  9,645  barrels 
in  February.  How  many  barrels  did  he  pack  in  the  three 
months  ?  S8, 096  larreU. 

84.  One  day  five  fishing-smacks  entered  the  harbor  of  Mar- 
blehead,  bringing  respectively  147  barrels  of  mackerel,  204 
barrels,  89  barrels,  246  barrels,  and  94  barrels.  How  many 
barrels  of  mackerel  did  all  of  them  bring  ?  780  barrels. 

85.  A  drover  paid  $5,897  for  465  head  of  cattle,  $3,486  for 
284  head,  $9,784  for  587  head,  and  $2,563  for  108  head. 
How  many  cattle  did  he  buy,  and  how  much  did  he  pay'  for 
them  ?  1, 444  head  of  cattle  ;  $21, 730, 

86.  A  merchant  buys  a  bale  of  sheeting,  containing  3  pieces 
of  38  yards  each,  4  pieces  of  39  yards  each,  6  pieces  of  42 
yards  each,  and  5  pieces  of  40  yards  each.  How  many  pieces 
in  the  bale  ?    How  many  yards  ?  722  yards. 

87.  At  the  battle  of  Gettysburg  the  loss  in  the  Union  army 
was  2,834  men  killed  and  13,790  wounded,  and  in  the  Confed- 
erate army  4,500  killed  and  26,500  wounded.  What  was  the 
whole  loss  in  each  army  ?    U?iion,  16,624;  Confederate,  31,000. 

88.  What  was  the  whole  number  of  men  killed  ?      7,334. 

89.  What  was  the  whole  number  wounded  ?  40,290. 

90.  What  was  the  whole  loss  in  both  armies  ?         4'^,^^4- 

91.  The  number  of  cattle  received  at  the  New  York  Cattle 
Market  in  one  week  was  226  by  the  New  York  and  Erie  Rail- 
road, 116  by  the  Hudson  River  Railroad,  2,669  by  the  Harlem 
Railroad,  319  by  the  New  Jersey  Central  Railroad,  445  by 
Hudson  River  boats,  and  26  on  foot.  How  many  cattle  were 
received  that  week  ?  3, 801. 

92.  The  value  of  the  gold  and  silver  exported  from  Califor- 
nia in  ten  years  commencing  with  1854,  was  as  follows : 


In  1854,  .  .$52,045,633 

In  1855,  .  .    45,161,731 

In  1856,  .  .    50,697,43^ 

In  1857,  .  .    48,976,697 

In  1858,  .  .    47,548,026 


In  1859, 
In  1860, 
In  1861, 
In  1862, 
In  1863, 


$47,640,462 
42,325,916 
40,676,758 
42,561,761 
46,071,920 


What  was  the  total  value  for  the  ten  years  ?    $463,706,338. 


SECTION   III. 

S  ITS  T^  ^CTIOJV, 


IN3DTJCTION. 

^  (See  Manual,  page  216.) 

24.  1.  Of  the  8  ladies  in  this  picture,  3  are  coming  down 
the  street,  and  the  others  are  going  up  the  street.  How  many- 
ladies  are  going  up  the  street  ? 

2.  Four  of  the  ladies  are  walking,  and  the  others  are  riding. 
How  many  are  riding  ? 

3.  In  the  picture  are  9  horses,  going  up  street,  and  the 
others  coming  down.  How  many  horses  are  coming  down  street  ? 

4.  All  but  3  of  the  9  horses  are  driven  in  teams.  How  many 
are  driven  in  teams  ? 

5.  There  are  12  barrels  in  the  picture,  5  of  them  on  a  cart, 
and  the  others  by  the  store  on  the  corner.  How  many  are  by 
the  store  ? 

6.  Of  the  13  men  shown  in  the  picture,  6  are  walking,  and 
the  others  are  riding.     How  many  are  riding  ? 

7.  Of  the  13  men,  10  are  coming  toward  us,  and  the  others 
are  going  from  ii§.     How  many  are  going  from  us  ? 


32  INTEGERS. 

25.  When  one  of  two  numbers  is  taken  from  iha 
other,  tlie  process  is  Subtraction. 

26.  The  result  thus  found  is  the  Remainder^  or  Dif- 
ference. 

27.  The  number  from  which  another  is  to  be  taken 
is  the  Minuend. 

28.  The  number  to  be  taken  from  another  is  the 
Subtrahend. 

The  number  of  ones  in  the  subtrahend  and  remain- 
der, taken  together,  must  equal  the  number  of  ones  in 
the  minuend. 

8.  Subtract  7  books  from  11  books. 

9.  What  will  be  the  remainder  if  you  take  9  chairs  from  16 
chairs  ? 

10.  If  6  cents  be  subtracted  from  15  cents,  what  will  be  the 
remainder  ? 

11.  A  cook,  having  18  eggs,  used  9  on  Monday,  and  the 
remainder  on  Tuesday.     How  many  did  she  use  on  Tuesday  ? 

13.  What  is  the  difference  between  17  leaves  and  8  leaves  ? 

13.  How  much  is  the  difference  between  14  bushels  of  pota- 
toes and  5  bushels  of  potatoes  ? 

14.  From  15  inches  subtract  7  inches. 

15.  Subtract  9  from  19. 

16.  The  minuend  is  13,  and  the  subtrahend  4.  What  is  the 
remainder  ? 

29.  This  sign  — ,  written  between  two  numbers,  sig- 
nifies that  the  number  after  it  is  to  be  subtracted  from 
the  number  before  it. 

It  is  called  Minus,  or  the  Sign  of  Subtraction.  Thus, 
25  —  16  —  9  is  read  25  minus  16  equals  9. 

17.  Read  15  -  7  =  8.  I     19.  Read  13+ 8  =  30  -  9. 

18.  Read  17-6  =  11.  I     20.  18  -  7  =  how  many  ? 
31.  31  brushes  —  11  brushes  =  how  many  brushes  ? 

33.  37  words  —  8  words  —  5  words  —  7  words  =  how  many 

words  ?  (Sec  Manual,  page  216.) 


SUBTRACTION. 


33 


30i     SUBTRACTION     TABLE. 


ojS 

12    3    4 
0    0    0    0 

5    6    7    8    9 
0    0    0    0    0 

-  j  5    6    7    8    9  10  11  12  13  14 
0\5    555555555 

0 

12    3    4 

5    6    7    8    9 

0123456789 

0 

2    3    4    5 
1111 
12    3    4 

6    7    §    910 
11111 
5    6    7    8    9 

^  (  6    7    8    9  10  11  12  13  14  15 

0|6    666666666 

0123456789 

2{| 

0 

3    4    5    6 
2    2    2    2 
12    3    4 

7    8    9  10  11 
2    2    2    2    2 
5    6    7    8    9 

- j  7    8    9  10  11  12  13  14  15  16 

7|7    777777777 

0123456789 

3J3 

4    5    6    7 
3    3    3    3 

8    9  10  11  12 
3    3    3    3    3 

o  j  8    9  10  11  12  13  14  15  16  17 
OJ8    888888888 

0 

12    3    4 

5    6    7    8    9 

0123456789 

4]t 

5    6    7    8 
4    4    4    4 

9  10  11  12  13 
4    4    4    4    4 

ft  j  9  10  11  12  13  14  15  16  17  18 
9]9    999999999 

0 

12    3    4 

5    6    7    8    9 

0123456789 

OBJLL    EXJEnCISES. 

1. — ^1,  Subtract  1  from  every  number  from  100  down  to  1,  thus ;  1 
from  100  leaves  99, 1  from  99  leaves  98, 1  from  98  leaves  97,  and  so  on. 
2.  Count  from  100  down  to  1,  thus ;  100,  99,  98,  97,  and  so  on. 

2. — 1.  Subtract  2  from  every  second  number  from  100  down  to  0, 
thus ;  2  from  100  leaves  98,  2  from  98  leaves  96,  2  from  96  leaves  94, 
and  so  on. 

2.  Count  by  2's  from  100  down  to  0,  thus ;  100,  98,  96,  94,  and  so  on. 

3.  Subtract  2  from  every  second  number  from  101  down  to  1,  thus ; 
2  from  101  leaves  99,  2  from  99  leaves  97,  2  from  97  leaves  95, 
and  so  on. 

4.  Count  by  2's  from  101  down  to  1,  thus ;  101, 99, 97,  95,  and  so  on. 

3. — 1.  Subtract  3  from  every  third  number  from  100  down  to  1, 
thus ;  3  from  100  leaves  97,  3  from  97  leaves  94,  and  so  on. 

2.  Count  by  3's  from  100  down  to  1,  thus ;  100,  97,  94,  91,  and  so  on. 

3.  Count  by  3's  from  101  down  to  2.  (See  Manual,  page  216) 

4.  .Count  by  3's  from  102  down  to  0. 

4. — 1.  Subtract  4  from  every  fourth  number  from  100  down  to  0, 
thus ;  4  from  100  leaves  96,  4  from  96  leaves  92,  4  from  92  leaves  88, 
and  so  on. 

2.  Count  by  4's  from  100  down  to  0,  thus ;  100, 96, 92, 88, 84,  and  so  on. 

3.  Count  by  4's  from  101  down  to  1. 

4.  Count  by  4's  from  102  down  to  2. 
6,  Count  by  4's  from  103  down  to  3. 


34  INTEGERS. 

5. — 1.  Subtract  5  from  every  fifth  number  from  100  down  to  0, 
thus ;  5  from  100  leaves  95,  5  from  95  leaves  90,  and  so  on. 

2.  Count  by  5's  from  100  down  to  0,  thus ;  100,  95,  90, 85,  and  so  on. 

3.  Count  by  5's  from  101  down  to  1. 

4.  Count  by  5'8  from  103  down  to  2. 

5.  Count  by  5's  from  103  down  to  3. 

6.  Count  by  5'8  from  104  down  to  4. 

6. — 1.  Subtract  6  from  every  sixth  number  from  102  down  to  0, 
thus ;  6  from  102  leaves  96,  6  from  96  leaves  90,  and  so  on. 

2.  Count  by  6's  from  102  down  to  0,  thus ;  102,  96,  90,  84,  and  so  on. 

3.  Count  by  6's  from  103  down  to  1. 

4.  Count  by  6's  from  104  down  to  2. 

5.  Count  by  6's  from  105  down  to  3. 

6.  Subtract  6  from  every  sixth  number  from  100  down  to  4. 

7.  Count  by  6's  from  101  down  to  5. 

7. — 1.  Subtract  7  from  every  seventh  number  from  105  down  to  0, 
thus ;  7  from  105  leaves  98,  7  from  98  leaves  91,  and  so  on. 

2.  Count  by  7's  from  105  down  to  0,  thus ;  105, 98, 91, 84,  and  so  on. 

3.  Count  by  7's  from  106  down  to  1. 

4.  Count  by  7's  from  100  down  to  2. 

5.  Subtract  7  from  every  seventh  number  from  101  down  to  3. 

6.  Subtract  7  from  every  seventh  number  from  102  down  to  4. 

7.  Count  by  7'8  from  103  down  to  5. 

8.  Count  by  7's  from  104  down  to  6. 

8. — 1.  Subtract  8  from  every  eighth  number  from  104  down  to  0, 
thus ;  8  from  104  leaves  96,  8  from  96  leaves  88,  and  so  on. 

2.  Count  by  8's  from  104  down  to  0,  thus ;  104,  96,  88,  80,  and  so  on. 

3.  Subtract  8  from  every  eighth  number  from  105  down  to  1. 

4.  Count  by  8's  from  106  down  to  2. 

5.  Count  by  8's  from  107  down  to  3. 

6.  Count  by  8's  from  100  down  to  4. 

7.  Subtract  8  from  every  eighth  number  from  101  down  to  5. 

8.  Count  by  8's  from  102  down  to  6. 

9.  Count  by  8's  from  103  down  to  7. 

9. — ^1.  Subtract  9  from  every  ninth  number  from  108  down  to  0, 
thus ;  9  from  108  leaves  99,  9  from  99  leaves  90,  and  so  on. 

2.  Count  by  9's  from  108  down  to  0,  thus ;  108,  99,  90,  81,  and  so  on. 

3.  Subtract  9  from  every  ninth  number  from  100  down  to  1. 

4.  Count  by  9's  from  101  down  to  2. 

5.  Count  by  9's  from  102  down  to  3. 

6.  Subtract  9  from  every  ninth  number  from  103  down  to  4. 

7.  Subtract  9  from  every  ninth  number  from  104  down  to  5. 

8.  Count  by  9's  from  105  down  to  6. 

9.  Count  by  9'8  from  106  down  to  7. 
10.  Count  by  9'8  from  107  down  to  8. 


SUBTRACTION.  35 

C^SE     I. 
No  figure  of  the  subtrahend  greater  than  the  corres- 
ponding figure  of  the  minuend. 

31.  Ex.  What  is  the  difference  between  8,397  and  3,265  ? 

Explanation. — Since  these  numbers  solution. 
are  too  large  to  be  subtracted  mentally,     8,397  Minuend. 
we  write    the   subtrahend  below  the    3,265  subtrahend. 
minuend,  with  the  ones  under  ones,  the    5,132  Difference. 
tens  under  tens,   the  hundreds  under 
hundreds^  and  the  thousands  under  thousands.     Com- 
menciQg  at  the  right,  we  take  the  5  ones  from  the  7 
ones,  and  the  remainder,  2  ones,  we  write  under  the 
ones.     We  next  take  the  6  tens  from  the  9  tens,  and  the 
remainder,  3  tens,  we  write  under  the  tens.     Then  2 
hundreds  from  3  hundreds  leave  1  hundred,  which  we 
write  under  the  hundreds;  and  3  thousands  from  8  thou- 
sands leave  5  thousands,  which  we  write  under  the  thou- 
sands.    The  result,  5,132,  is  the  difference  or  remainder 
required. 

32.  We  can  subtract  apples  from  apples,  dollars  from 
dollars,  pens  from  pens,  or  hours  from  hours  ;  but  we 
can  not  subtract  apples  from  dollars,  nor  pens  from 
hours.  For  13  apples  —  4  dollars  =  neither  9  apples 
nor  9  dollars. 

Again,  we  can  subtract  ones  from  ones,  tens  from 
tens,  or  hundreds  from  hundreds  ;  but  we  can  not  sub- 
tract ones  from  hundreds,  nor  tens  from  thousands. 
For  9  thousands  —  4  tens  =  neither  5  tens  nor  5 
thousands.     Hence, 

33t   General   'PHnciptes  of  Subtraction, 

I.  Only  numbers  expressing  the  same  kind  of  things  can 
he  subtracted  the  one  from  the  other. 

n.  Only  figures  occupying  the  same  place  in  different 
numbers  can  be  subtracted  the  onefrmn,  the  other. 

(See  Manual,  page  216.) 


^0 

INT 

EGERS. 

rMOBJLEMS. 

(1) 

62 
41 

(2) 
76 

24 

(3) 
45 
34 

(4) 
57 
43 

(5)                 (6) 

682                584 
850                302 

(7) 

635  pins 
412  pins 

(8) 
3,846  soldiers 
2,534  soldiers 

(9) 
7,968  shingles 
5,453  shingles 

(10) 
57,908  pounds 
43,700  pounds 

11.  James,  having  27  marbles,  gave  12  of  them  to  John. 
How  many  marbles  had  he  left  ?  15. 

12.  From  a  piece  of  muslin  containing  39  yards,  a  merchant 
sold  13  yards  for  a  dress.  How  many  yards  remained  in  the 
piece  ?  26. 

13.  Joseph  had  46  cents,  but  he  has  spent  25  cents  for  a 
knife.     How  many  cents  has  he  now  ? 

14.  Ellen  attended  school  63  days  in  a  term  of  75  school- 
days.    How  many  days  was  she  absent  ?  12. 

15.  A  gardener  picked  68  boxes  of  strawberries  one  fore- 
noon, and  54  boxes  in  the  afternoon.  How  many  more  boxes 
did  he  pick  in  the  forenoon  than  in  the  afternoon  ? 

16.  Hiram  lives  98  rods  from  the  schoolhouse,  and  Thomas 
41  rods.  How  much  farther  does  Hiram  walk  in  going  to 
school  than  Thomas  ?  57  rods. 

17.  435  miles  —  314  miles  =  how  many  miles  ?  121. 

18.  6,798  bushels  —  2,641  bushels  =  how  many  bushels? 

19.  How  many  tons  are  38,156  tons  —  14,044  tons  ?    2^,112. 

20.  A  fruit-dealer,  having  247  baskets  of  peaches,  sold  125 
baskets.     How  many  baskets  had  he  left  ?  122. 

21.  A  man  whose  income  is  $875  a  year,  expends  $734. 
How  much  money  does  he  save  ?  $lJi'l. 

22.  A  drover  bought  a  lot  of  cattle  for  $4,574,  and  sold  them 
for  $5,896.     How  much  did  he  gain  ?  $1,322. 

(23)  (24)  (25)  (26) 

57,698  675,004  2,174,943  167,065,149 

43,257  245,002  42,301  4,042,136 

27.  One  year  a  farmer  raised  1,898  bushels  of  oats,  and  sold 
1,427  bushels.    How  many  bushels  did  he  keep  for  use  ?    J!i71. 


SUBTRACTION, 


OA.SE     II. 


37 


Any  figure  of  the  subtrahend  greater  than  the  corres- 
ponding figure  of  the  minuend. 

FIRST     METHOD. 

31.  Ex.  1.  From  16  subtract  9. 

(See  Manual,  page  216.) 

Explanation. — ^We  write  the  mimbers  as  in  solution. 

Case  I ;  but  as  we  can  not  subtract  9  ones  16 

from  6  ones,  we  must  unite  the  1  ten,  wbich  _9 

equals  10  ones,  with  the  6  ones,  and  subtract  7 
the  9  from  the  whole  16  at  once. 

Ex.  2.  From  76  subtract  29. 
Explanation. — ^As  we  can  not  subtract  9  ones    solution. 
from  6  ones,  we  take  1  of  the  7  tens  and  unite       — 
it  with  the  6  ones,  making  16  ones,  and  sub-       o  9 
tr acting  9  ones  from  the  16  ones,  we  write  the       t-z 
remainder,  7,  as  the  ones  of  the  final  result. 
Since  we  have  already  used  one  of  the  7  tens,  we  have 
now  only  6  tens  in  the  minuend,  and  hence  we  subtract 
the  two  tens  from  6  tens,  and  write  the  remainder,  4 
tens,  as  the  tens  of  the  final  result. 

:pnonjLEM8. 

28.  In  a  public  school  are  45  pupils,  and  28  of  them  are 
girls.     How  many  are  boys  ?  17. 

29.  A  groceV  sold  35  bars  of  soap  from  a  box  that  contained 
64  bars.    How  many  bars  were  left  in  the  box  ?  29. 

30.  One  day  92  boats  passed  Lockport  on  the  Erie  Canal, 
and  47  of  them  were  going  east.    How  many  were  going  west  ? 

31.  A  washerwoman  had  72  clothes-pins,  but  she  has  lost  29 
of  them.     How  many  has  she  now  ?  JiS. 

32.  A  jar  filled  with  butter  weighs  52  pounds,  and  the  jar 
alone  weighs  15  pounds.    How  much  does  the  butter  weigh  ? 


38  INTEGERS. 

33.  From  a  barrel  of  sugar  containing  283  pounds,  a  mer- 
cliant  sold  156  pounds.  How  many  pounds  were  left  in  the 
barrel  ?  121. 

34.  A  man  bought  a  village  lot  for  |350,  and  paid  down  all 
but  $125.     How  much  did  he  pay  ? 

35.  A  man  bought  a  piano  for  |475.  He  paid  $267  in  cash, 
and  gave  his  note  for  the  balance.  For  what  sum  did  he  give 
his  note  ?  $208. 

35.  Ex.  From  853  subtract  467. 

Explanation. — As  we   can   not    subtract   7  solution. 
ones  from  3  ones,  we   take  1  of  the  5  tens    ^-^ 
and  unite  it  with  the  3  ones,  making  13  ones ;    ^  g  y 
and  subtracting  7  ones  from  13  ones,  we  write    — — — 
the  remainder,  6,  as  the  ones  of  the  final  re- 
sult.    Since  we  have  already  used  1  of  the  5  tens,  only 
4  tens  now  remain  in  the  minuend.     As  we  can  not 
subtract  the  6  tens  from  4  tens,  we  unite  1  of  the  8  hun- 
dreds with  the  4  tens,  making  14  tens  ;  then  subtract- 
ing 6  tens  from  14  tens,  we  write  the  remainder,  8  tens, 
as  the  tens  of  the  final  result.     Since  we  have  already 
used  1  of  the  8  hundreds,  only  7  hundreds  now  remain 
in  the  minuend  ;  and  from  this  we  subtract  the  4  hun- 
dreds, and  write  the  remainder,  3  hundreds,  as  the 
hundreds  of  the  final  result. 

FMOBIj  EMS. 

36.  A  man  who  had  a  farm  of  154  acres,  gave  to  his  son  65 
acres.     How  much  land  had  he  left  ?  89  acres. 

37.  In  a  certain  village  school-district  are  447  children,  of 
whom  only  298  attend  school.  How  many  do  not  attend 
school  ?  1J^9  children. 

38.  A  provision  dealer  receiving  ah  order  for  525  barrels  of 
beef,  has  only  354  barrels  on  hand.  How  many  barrels  more 
will  he  require  to  fill  the  order  ? 

39.  A  merchant's  sales  in  January  amounted  to  $1743,  and 
in  February  to  $928.  How  much  did  the  sales  of  January  ex- 
ceed those  of  February  ?  $815. 


SUBTRACTION".  39 

40.  In  a  certain  town  135  men  were  drafted  for  tlie  army, 
but  54  of  them  were  rejected  by  the  examining  surgeon.  How 
many  passed  examination  ? 

41.  A  regiment  entered  the  service  with  1,149  men,  and  at 
the  close  of  the  war  had  only  437.  How  many  men  had  it 
lost  ?  '  ^722. 

42.  A  banker's  income  last  year  was  $12,849,  and  his  ex- 
penses were  $6,768.  How  much  did  his  income  exceed  his 
expenses?  $6,081. 

43.  A  market-gardener  in  one  year  received  $3,730  for  fruits 
and  vegetables,  and  his  expenses  were  $1,850.  How  much 
were  his  profits  ?  $1,880. 

44.  One  day  724  cattle  were  received  at  the  Philadelphia 
Cattle  Market,  and  648  of  them  were  sold.  How  many  re- 
mained unsold  ? 

45.  A  man  having  $974  in  the  bank,  drew  out  $396.  How 
much  money  had  he  left  on  deposit  ?  $578. 

36.  Ex.  From  3000  subtract  57. 

Explanation. — ^We  can  not  subtract  7  ones    solution. 
from  0  ones,  and  as  we  have  in  the  minuend     „ 
0  tens  to  unite  with  the  ones,  and  0  hundreds  g  y 

to  unite  with  the  tens,  we  must  take  1  of  the  3    

thousands,  leaving  2  thousands.  This  1  thou-  ^^^^ 
sand  =  10  hundreds  ;  but  as  we  can  not  subtract  ones 
from  hundreds  (Prin.  11.),  we  take  1  of  the  10  hun- 
dreds, leaving  9  hundreds.  This  1  hundred  =  10  tens ; 
but  as  we  can  not  subtract  ones  from  tens,  we  take  1 
of  the  10  tens,  leaving  9  tens.  This  1  ten  =  10  ones. 
We  now  take  the  7  ones  from  10  ones,  and  the  remain- 
der, 3  ones,  we  write  as  the  ones  of  the  final  result. 
Then  5  tens  from  9  tens  leave  4  tens,  and  as  there  are  no 
hundreds  or  thousands  in  the  subtrahend,  we  write  the 
9  hundreds  and  the  2  thousands  of  the  minuend,  for 
the  hundreds  and  thousands  of  the  final  result. 


iy^^ZZ^ 


40  INTEGERS. 


JPJROBJLEMS. 


46.  If  I  buy  a  bushel  of  apples  for  65  cents,  and  give  in 
payment  a  dollar  bill,  how  much  change  should  I  receive  ? 

47.  At  a  flouring-mill  in  Baltimore  1000  barrels  of  flour 
were  made  in  one  week,  and  869  barrels  of  it  were  sold.  How 
many  barrels  were  unsold  ?  isi, 

48.  The  Phillips  Well  on  Oil  Creek  is  460  feet  deep,  and 
the  Titusville  Well  1100  feet  deep.  How  much  deeper  is  the 
latter  well  than  the  former  ?  SIfifeet. 

49.  A  man  divided  $7,500  between  his  son  and  daughter, 
giving  $4,375  to  his  son.  How  much  did  the  daughter  re- 
ceive? $3,225. 

50.  A  shipbuilder  received  $21,000  for  a  schooner,  which 
cost  him  $18,728.     How  much  was  his  gain  ?  $2,272. 

51.  A  forwarder  had  40,000  bushels  of  oats  in  store  at  Chi- 
cago. How  many  bushels  had  he  in  store  after  shipping 
25,487  bushels  to  Buffalo  ?  U,513  ImsheU. 

52.  A  broker  sold  stocks  for  $256,200  which  cost  him 
$209,408.  Did  he  gam  or  lose,  and  how  much  ?   Gained  $46, 792. 

53.  How  many  acres  are  1,100  acres  —  841  acres  ?      259. 

54.  How  many  cords  are  21,610  cords  —  19,587  cords  ? 

55.  How  many  gallons  are  110,040  gallons— 90,621  gallons? 

56.  At  the  battle  of  Bunker  Hill  the  Americans  lost  449 
men,  and  the  British  1054.  How  much  did  the  British  loss 
exceed  the  American  ? 

SECOND    METHOD. 

87.  Ex.  rrom  7,623  subtract  4,856. 

Explanation.  —  In    this    solution  we  com-  ^^J:^^^' 
mence  at  the  right,  and  proceed  the  same  as     I'^r^ 
in  the  First  Method,  except  that  we  omit  to    — — 
write  the  partial  minuends,  13  ones,  11  tens,     2,767 
15  hundreds,  and  6  thousands,  above  the  given  min- 
uend. (See  Manual,  page  216.) 


6UBTKAOTION.  41 

38.  "Rule  for  Subtraction  of  Integers, 

I.  Write  the  subtrahend  below  the  minuend^  placing  ones 
under  ones,  tens  under  tens,  and  so  on. 

n.  When  the  figures  of  the  subtrahend  do  not  ex- 
ceed in  value  the  corresponding  figures  of  the  minuend, 

1.  Commencing  at  the  right  hand,  subtract  each  figure 
of  the  subtrahend  from  the  corresponding  figure  of  the 
minuend,  and  write  the  result  directly  below  in  the  re- 
mainder. 

2.  If  there  are  figures  in  the  minuend  unthout  any  cor- 
responding figures  in  the  subtrahend,  write  them  m  the 
remainder. 

m.  When  any  figure  of  the  subtrahend  exceeds  the 
corresponding  figure  of  the  minuend, 

Add  10  to  the  figure  of  the  minuend,  and  from  the  sum 
subtract  the  figure  of  the  subtrahend.  In  this  case,  always 
call  the  next  left-hand  figure  of  the  minuend  1  less,  or  the 
next  left-hand  figure  of  the  subtrahend  1  more. 

(See  Manual,  page  216.) 
JPM  OBJOEMS, 

57.  A  load  of  hay  with  the  wagon  weighed  2,656  pounds 
on  the  hay  scales,  and  the  wagon  alone  weighed  987  pounds. 
How  much  did  the  hay  weigh  ?  1, 669  pounds. 

58.  At  a  certain  election  2,649  votes  were  cast  for  one  can- 
didate, and  1,975  votes  for  the  other.  What  majority  did  the 
successful  candidate  receive  ?  674  'ootes. 

(59)  (60)  (61)  (62) 

1,000,000  348,794  7,408,215  300,300,333 

31,276  127,586  59,826  47,008,296 

63.  The  island  of  Cuba  contains  45,277  square  miles,  and 
the  State  of  Ohio  39,904  square  miles.  How  much  larger  is 
Cuba  than  Ohio  ? 

D 


4:2 


INTEGERS 


The  numbers  on  this  map  of  Mis- 
sissippi River  show  the  distances 
of  the  different  places  from  the 
mouth  of  the  river. 

How  many  miles  is  it  from 

64.  Eock  Island  to  Yicksburg  ? 

65.  Memphis  to  La  Crosse ?  1,0 JfS. 

66.  Cairo  to  Lake  Itasca  ?     1,331. 
77.  St.  Paul  to  Baton  Rouge  ? 

68.  The  Falls  of  St.  Anthony  to  the 

mouth  of  Red  River  ?  1, 836. 

69.  Burlington  to  Natchez  ?  i.j^i^. 

70.  The  mouth  of  Missouri  River 

to  Prairie  Du  Chien  ?     538. 

71.  New  Orleans  to  Quincy  ?  1, 399. 
73.  Quincy  to  St.  Paul  ?  632. 

73.  The  mouth  of  Arkansas  River 

to  the  mouth   of  Missouri 
River  ?  64S. 

74.  Keokuk  to  New  Orleans  ? 
95.  Du  Buque  to  Lake  Itasca  ? 

76.  Prairie  Du  Chien  to  Natchez? 

77.  St.  Louis  to  the  Falls  of  St. 

Anthony  ?  798. 

78.  Memphis  to  Baton  Rouge  ? 

79.  Keokuk  to  La  Crosse?        4I8. 

80.  Du  Buque  to  Vicksburg  ? 

81.  Cairo  to   the  mouth  of  Red 

River  ?  850. 

83.  The  mouth  of  Arkansas  River 

to  Lake  Itasca ?  1,769. 

83.  St.  Louis  to  Rock  Island  ? 

84.  Burlington  to  Memphis  ? 

85.  St.  Paul  to  Vicksburg  ?  1,630. 

86.  Quincy  to  Lake  Itasca  ?     985, 

87.  New  Orleans  to  the  Falls  of 

St.  Anthony?  2,039. 

88.  Rock  Island  to  Baton  Rouge  ? 

89.  St.  Louis  to  New  Orleans? 


SUBTRACTION.  43 

90.  If  I  owe  $3,496,  and  I  pay  $1,748,  how  much  do  I  then 
owe?  $l,7Ji8. 

91.  What  is  the  difference  between  9,417  and  3,584  ?     5, 833. 
93.  The  greater  of  two  numbers  is  11,419,  and  the  less  is 

7,255.     What  is  their  difference  ?  ^  16Jf. 

93.  The  running  expenses  of  a  machine-shop  for  a  year 
were  $30,456,  and  the  sales  amounted  to  $31,317.  What  were 
the  net  earnings  for  the  year  ?  $10,761. 


MEVIJEW   PJROBIjEMS. 

1.  In  the  year  1860  there  were  34  States  in  the  Union,  of 
which  11  seceded.     How  many  States  did  not  secede  ?     23. 

3.  A  farmer  drew  to  market  seven  loads  of  hay,  which 
weighed  1,577  pounds,  1,891  pounds,  1,648  pounds,  3,154 
pounds,  1,736  pounds,  1,954  pounds,  and  2,036  pounds.  What 
was  the  weight  of  the  seven  loads  ? 

3.  On  board  an  ocean  steamer  were  114  cabin  passengers, 
649  steerage  passengers,  and  a  crew  of  87  persons.  What  was 
the  whole  number  of  persons  on  board  ?  850. 

4.  A  father  and  his  two  sons  earned  $1875  in  a  year,  the 
elder  son  earning  $638,  and  the  younger  son  $459.  How 
much  did  the  father  earn  ?  $778. 

5.  A  gardener  received  $318  for  cabbages,  and  $439  for 
tomatoes.  The  expense  of  raising  the  cabbages  was  $84,  and 
of  the  tomatoes  $134.  What  were  his  profits  on  the  two 
crops  ?  $U9. 

6.  A  regiment  when  it  entered  the  service  mustered  1004 
men;  during  the  war  37  of  these  were  killed,  48  died,  53 
were  taken  prisoners,  and  597  were  discharged.  How  many 
men  served  through  their  term  of  enlistment  ?  269. 

7.  In  five  successive  weeks  79,747  tons,  84,334  tons,  68,953 
tons,  76,081  tons,  and  81,168  tons  of  coal  were  taken  to  Phil- 
adelpliia  by  the  Philadelphia  and  Reading  Eailroad.  How 
many  tons  were  carried  over  the  road  in  the  five  weeks  ? 

8.  A  contractor  furnished  10,000  overcoats  for  the  army,  but 
1,715  of  them  were  condemned  as  imperfect.  How  many  of 
them  were  accepted  ? 


44  INTEGERS.. 

9.  On  the  first  of  January  an  edition  of  5,000  copies  of  a 
book  was  published.  In  January  396  copies  were  sold,  in 
February  741,  in  March  1,214,  in  April  927,  in  May  643,  and 
in  June  584.    How  many  copies  remained  unsold  July  1  ?    J,.95. 

10.  A  man  was  bom  in  the  year  1799,  and  died  at  the  age  of 
07  years.     In  what  year  did  he  die  ?  In  the  year  18G6. 

11.  The  battle  of  Lexington  was  fought  in  the  year  1775,  and 
President  Lincoln  was  assassinated  in  1865.  How  many  years 
between  the  two  events  ?  90. 

12.  In  the  year  1861  the  senior  class  of  a  certain  college 
contained  46  students,  the  junior  class  38,  the  sophomore 
class  59,  and  the  freshman  class  74.  Of  these,  12  seniors,  14 
juniors,  18  sophomores,  and  25  freshmen  enlisted.  How  many 
students  enlisted  ? 

13.  How  many  students  remained  in  each  class  ?  How  many 
remained  in  college  ?  I48  reTnained  in  college. 

14.  A  railroad  train  left  Cincmnati  for  St.  Louis  with  435 
passengers.  On  the  route  215  passengers  left  the  cars,  and 
194  went  aboard.  How  many  were  on  the  train  when  it 
reached  St.  Louis  ?  Jf,lJ^. 

15.  One  year  a  merchant's  sales  amounted  to  |37,496.  His 
goods  cost  $25,267,  and  his  store  expenses  were  $6,485.  How 
much  were  his  profits  ?  $5,  IJ^J^. 

16.  One  day  48,325  letters  were  received  at  the  New  York 
Post-office.  Of  these,  21,259  were  directed  to  places  m  the 
State  of  New  York,  20,048  to  places  in  other  states,  and  the 
rest  to  places  in  foreign  countries.  How  many  were  directed 
to  foreign  countries  ?  7,018. 

-  17.  At  the  beginning  of  the  year  A  had  property  worth 
$2,350,  but  he  owed  $476.  During  the  year  he  earned  $1,156, 
and  expended  $879.  How  much  was  he  worth  at  the  end 
of  the  year?  $2,151, 

18.  1,153  —  967  +  10,000  —  5,308  =  how  many? 

19.  What  is  the  difierence  between  13,647  +  2,593  —  6,483 
4,931  +  5,006  -  7,285  ?  3,033. 


MULTIPLICATION. 


45 


SECTION    IV. 

MZrZ  TITZIC;A  TIOJV. 

IK-DXJCTION". 

(See  Manual,  page  216.) 

30«  1.  In  this  picture  are  3  stems  of  a  rose-bush,  with  3 
ro&es  on  each  stem.  3  roses  and  3  roses  and  3  roses  are  how 
many  roses  ?     Then,  3  times  3  roses  are  how  many  roses  ? 

2.  On  each  of  the  stems  are  4  buds.  4  buds  and  4  buds  and 
4  buds  are  how  many  buds  ?     3  times  4  buds  are  how  many  ? 

3.  On  the  branches  of  a  cherry-tree  we  see  5  clusters,  with 
3  cherries  in  each  cluster.  3  cherries  -f  3  cherries  +  3  cherries 
+  3  cherries  4-  3  cherries  are  how  many  cherries  ?  5  times  3 
cherries  are  how  many  ? 

4.  On  each  of  the  5  branches  are  4  leaves.  5  times  4  leaves 
are  how  many  leaves  ? 

5.  A  boy  at  play  with  his  blocks  places  them  in  rows  on 
a  table.  If  he  counts  them  one  way  he  has  3  rows,  and  7 
blocks  in  each  row ;  but  if  he  counts  them  another  way  he 
has  7  rows,  and  3  blocks  in  each  row.  How  many  blocks 
has  he  ? 

How  many  are 

6.  3  times  7  blocks  ?  |  7.  7  times  3  blocks  ? 

8.  If  a  woman  can  weave  6  yards  of  rag  carpet  in  1  day, 
how  many  yards  can  she  weave  in  4  days  ? 


46  INTEGERS. 

9.  A  shoemaker  can  make  5  pairs  of  children's  shoes  in  1 
day.    How  many  pairs  can  he  make  in  8  days  ? 

10.  How  much  will  5  oranges  cost,  at  8  cents  a  piece  ? 

The  sum  of  7  +  7  +  7  +  7  +  7  is  35,  and  5  times 
7  are  35.  By  each  of  these  methods  we  have  fotuid  the 
sum  of  five  7's,  or  of  as  many  7's  as  there  are  ones  in  5  ; 
but  the  second  method  is  shorter  than  the  first. 

40.  The  process  of  finding,  by  a  method  shorter  than 
Addition,  the  sum  of  as  many  times  one  number  as 
there  are  ones  in  another,  is  Multiplication. 

41.  The  result  thus  found  is  the  Product,  and 

42.  The  numbers  themselves  are  Factors. 

43.  The  factor  which  is  to  be  taken  any  certain  num- 
ber of  times  is  the  Multiplicand,  and 

44.  The  factor  which  shows  how  many  times  the  mul- 
tiplicand is  to  be  taken,  is  the  Multiplier. 

The  number  of  ones  in  the  product  must  equal  the 
number  of  ones  of  the  multiplicand  taken  as  many 
times  as  there  are  ones  in  the  multipher. 

11.  What  is  the  product  of  6  times  8  loaves  of  bread  ? 

12.  What  is  the  product  of  the  factors  10  and  7  ? 

13.  7  times  8  are  56.  Which  of  these  numbers  is  the  mul- 
tiplicand ?  Which  is  the  multiplier  ?  Which  is  the  product  ? 
Which  are  the  factors  ? 

45.  This  sign  x,  written  between  two  numbers,  signi- 
fies that  they  are  to  be  multiplied  together. 

It  is  called  the  Sign  of  Multiplication,  and  is  read 
times,  or  multiplied  hy.     Thus, 

6  X  9  =  54  may  be  read  6  times  9  equal  54,  6  times  9 
are  54,  6  multiplied  by  9  equal  54,  or  6  multiplied  by  9 
are  64. 

14.  Read  7  x  12  =  84.  16.  Read  3x8  =  4x6. 

15.  Read  4  x  25  =  100.  17.  Read  3  x  4  x  5  =  60. 
18.  6  X  7  plums  are  how  many  plums? 


MULTIPLICATION. 


47 


19.  10  X  4  apples  =  how  many  apples  ? 

20.  3  X  2  X  5  balls  =  how  many  balls  ? 

,46.  Numbers  applied  to  objects  or  things  are  Concrete 
Numbers;  as,  4  apples,  19  men,  237  books. 

47f  Numbers  not  applied  to  objects  or  things  are 
Abstract  Numbers;  as,  4,  19,  237.  (See  Manual,  page  216.) 

48.    MULTIPLICATION    TABLE. 


1]? 

1    2 
1    1 

3    4    5    6    7    8    9 
1111111 

oJO    123456789 
0|6    666666666 

0 

1    2 

3    4    5    6    7    8    9 

0    6  12  18  34  30  36  42  48  54 

o(0 

1  2 

2  2 

3    4    5    6    7    8    9 
2    2    2    2    2    2    2 

„jO    123456789 

7|7    777777777 

0 

2    4 

6    8  10  12  14  16  18 

0    7  14  21  28  35  43  49  56  63 

3\t 

1    2 
3    3 

3    4    5    6    7    8    9 
3    3    3    3    3    3    3 

QJO    123456789 

0I8    888888888 

0 

3    6 

9  12  15  18  21  24  27 

0    8  16  24  33  40  48  56  64  72 

412 

1    2 
4    4 

3  4    5    6    7    8    9 

4  4    4    4    4    4    4 

ftjO    123456789 
9]9    999999999 

0 

4    8  13  16  20  34  28  32  36 

0    9  18  27  36  45  54  63  73  81 

0 

123456789 
555555555 
5  10  15  20  35  30  35  40  45 

irt(  0133456789 
lUjio  10  10  10  10  10  10  10  10  10 

0  10  20  30  40  50  60  70  80  90 

OUAIj     XJXJSJtCISES. 

1. — 1.  Count  by  3's  to  20,  in  this  manner :  One  2  is  2,  or  once  2  is 
2 ;  two  3'8  are  4,  or  3  times  2  are  4 ;  three  2's  are  6,  or  3  times  2  are  6 ; 
four  3^s  are  8,  or  4  times  3  are  8;  and  so  on. 

3.  Multiply  from  0  times  3  to  10  times  3,  thus ;  0  times  2  is  0,  once 

2  is  3,  3  times  3  are  4,  3  times  3  are  6,  4  times  3  are  8,  and  so  on. 

3.  Multiply  from  10  times  2  to  0  times  2,  thus ;  10  times  2  are  20, 
9  times  3  are  18,  8  times  3  are  16,  7  times  3  are  14,  and  so  on. 

2. — 1.  Count  by  3's  to  30,  in  this  manner :  One  3  is  3,  or  once  3  is  3 ; 
two  3's  are  6,  or  3  times  3  are  6;  three  3's  are  9,  or  3  times  3  are  9; 
four  3's  are  13,  or  4  times  3  are  12,  and  so  on. 

2.  Multiply  from  0  times  3  to  10  times  3,  thus ;  0  times  3  is  0,  once 

3  is  3,  2  times  3  are  6,  3  times  3  are  9,  and  so  on. 

3.  Multiply  from  10  times  3  to  0  times  3,  thus  ;  10  times  3  are  30, 
9  times  3  are  27,  8  times  3  are  24,  and  so  on. 

3. — 1.  Count  by  4's  to  40,  thus :  One  4  is  4,  or  once  4  is  4;  two  4's 
are  8,  or  2  times  4  are  8 ;  three  4's  are  13,  or  3  times  4  are  13,  and  so  on. 


48  INTEGERS. 

2.  Multiply  from  0  times  4  to  10  times  4,  thus ;  0  times  4  is  0 
once  4  is  4,  2  times  4  are  8,  3  times  4  are  12,  and  so  on. 

3.  Multiply  from  10  times  4  to  0  times  4,  thus ;  10  times  4  are  40, 
9  times  4  are  36,  8  times  4  are  32,  and  so  on. 

4. — 1.  Count  by  5'8  to  50.     (See  Manual,  page  216.) 

2.  Multiply  from  0  times  5  to  10  times  5. 

3.  Multiply  from  10  times  5  to  0  times  5. 
6.— 1.  Count  by  6's  to  60. 

2.  Multiply  from  0  times  6  to  10  times  6. 

3.  Multiply  from  10  times  6  to  0  times  6. 
6.— 1.  Count  by  7's  to  70. 

2.  Multiply  from  0  times  7  to  10  times  7. 

3.  Multiply  from  10  times  7  to  0  times  7. 
7.— 1.  Count  by  8's  to  80. 

2.  Multiply  from  0  times  8  to  10  times  8. 

3.  Multiply  from  10  times  8  to  0  times  8. 
8.— 1.  Count  by  9's  to  90. 

2.  Multiply  from  0  times  9  to  10  times  9. 

3.  Multiply  from  10  times  9  to  0  times  9. 
9.— 1.  Count  by  lO's  to  100. 

2.  Multiply  from  0  times  10  to  10  times  10. 

3.  Multiply  from  10  times  10  to  0  times  10. 

C  u^  S  E     I . 
The  Multiplier  One  Figure. 

FIRST    METHOD. 

49.  Ex.  1.  37  +  37  +  37  +  37,  or  4  times  37,  are  how 

Ulciliy  .  FIRST  SOLUTION. 

Explanation. — ^In  adding  these  numbers,     s^  Addition. 
we  first  find  the  sum  of  7  ones  taken  4  37 

times,  which  is  28.     We  write  the  8  ones  37 

of  this  sum  as  the  ones  of  the  final  result,  ^|^ 

and  the  2  tens  we  write  in  tens'  place  be-  — 

tween  the  two  parallel  lines.    We  next  find        _2_ 
the  sum  of  3  tens  taken  4  times,  which        148 
is  12  tens  ;   and  adding  to  this  sum  the  2 
tens  of  the  first  result,  we  write  the  14  tens,  which 
equal  4  tens  and  1  hundred,  as  the  tens  and  hundreds 
of  the  final  result. 


MULTIPLICATION.  49 

In  the  second  solution  we  write  the  37    «=^<^^  solution. 

_  T  J 1  •  1  •      J       1        J3u  Multiplication. 

only  once,  and  as  this  number  is  to  be  07 

taken  4  times,  we  write  4  under  the  ^ 

right-hand  figure.     7  ones  +  7  ones  +  7  —z- 

ones  +  7  ones,  or  4  times  7  ones,  are  28  

ones.     We  write  the  8  ones  of  this  sum  ^^^ 

or  product  as  the  ones  of  the  final  result ;  and  the  2 
tens,  which  are  to  be  added  to  the  sum  or  product  of 
the  tens,  we  write  in  tens'  place  between  the  two  paral- 
lel Hues.  Then,  3  tens  +  3  tens  +  3  tens  +  3  tens,  or  4 
times  3  tens,  are  12  tens  ;  and  adding  to  this  sum  or 
product  the  two  tens  of  the  first  result,  we  write  the  14 
tens  as  the  tens  and  hundreds  of  the  final  result. 

(See  Manual,  page  216.) 

Ex.  2.  What  is  the  product  of  3,794  multipHed  by  6  ? 
Explanation. — ^We  write  the  mul-      solution. 
tipher  under  the  multipHcand,  and        ^^94  Multiplicand. 

commence  at  the  right  to  multiply.         ^  ^^  ^^^' 

6x4  ones  =  24  ones,  or  4  ones  and  452 
2  tens.  We  write  the  4  ones  for  the  22764  Product 
ones  of  the  final  product,  and  the  2 
tens  in  tens'  place  between  the  parallel  Hues.  6x9 
tens  =  54  tens,  and  54  tens  +  2  tens  =  56  tens,  or  6 
tens  and  5  hundreds.  We  write  the  6  tens  for  the  tens 
of  the  final  product,  and  the  5  hundreds  in  hundreds' 
place  between  the  parallel  lines.  6x7  hundreds  =  42 
hundreds,  and  42  hundreds  +  5  hundreds  =  47  hun- 
dreds, or  7  hundreds  and  4  thousands.  We  write  the 
7  hundreds  for  the  hundreds  of  the  final  product,  and 
the  4  thousands  in  thousands'  place  between  the  paral- 
lel lines.  6x3  thousands  =  18  thousands,  and  18 
thousands  +  4  thousands  =  22  thousands,  or  2  thou- 
sands and  2  ten-thousands,  and  these  we  write  for  the 
thousands  and  ten-thousands  of  the  final  product. 

E 


50  INTEGERS. 


PJROBIjJEMS, 


1.  How  much  will  3  tons  of  hay  cost,  at  $12  a  ton  ?     $36. 

2.  In  1  day  there  are  24  hours.     How  many  hours  in  4 
?  96. 

3.  How  much  will  4  pounds  of  raisins  cost,  at  22  cents  a 
pound  ?  88  cents. 

4.  If  a  canal-boat  goes  42  miles  in  a  day,  how  many  miles 
will  it  go  iQ  6  days  ?  252. 

5.  A  farm  laborer  worked  5  months  for  $16  a  month. 
How  much  did  his  wages  amount  to  ?  $80. 

6.  What  is  the  product  of  7  x  74  ?  518. 

7.  The  multiplicand  is  96,  and  the  multiplier  is  8.  What 
is  the  product  ?  768. 

8.  A  livery-man  bought  3  horses,  at  $125  apiece.  How 
much  did  they  cost  him  ? 

9.  How  much  will  it  cost  for  a  party  of  6  persons  to  go 
from  New  York  to  Liverpool  on  an  ocean  steamer,  if  the  fare 
is  $137  ?  $822. 

10.  A  music  dealer  sold  7  pianos,  at  $325  each.  How  much 
did  he  receive  for  all  of  them  ?  $2,275. 

11.  How  many  quarts  of  milk  will  be  used  in  a  hotel  in  a 
week,  if  36  quarts  are  used  each  day  ? 

12.  The  factors  are  2,147  and  5.    What  is  the  product  ? 

13.  A  cooper  sent  to  the  mill  9  loads  of  flour-barrels,  and 
each  load  contained  146  barrels.  How  many  barrels  did  he 
send? 

14.  In  one  mile  there  are  5,280  feet.  How  many  feet  in  8 
miles  ? 

15.  The  Pennsylvania  Central  Kailroad  Company  bought  6 
locomotives,  at  $28,675  each.     How  much  did  they  all  cost  ? 

$172,050. 

16.  How  much  will  7  bushels  of  potatoes  cost,  at  56  cents  a 
bushel  ?  S92  cents. 

17.  A  farmer  raised  8  acres  of  wheat,  and  each  acre  pro- 
duced 25  bushels.    How  many  bushels  of  wheat  did  he  raise  ? 


MULTIPLICATION.  61 

SECOND    METHOD. 

50.  Ex.  Multiply  473  by  9. 

Explanation. — ^We  multiply  each  figure  of  solution. 
tlie  multiplicand  by  the  multiplier,  as  in  the      473 

First  Method.     9x3  ones  are  27  ones,  or  7    9 

ones  and  2  tens.  We  write  the  7  ones  as  the  4,257 
ones  of  the  product,  but  instead  of  writing 
down  the  tens' figure,  2,  we  reserve  it  in  the  mind,  to  be 
added  to  the  product  of  the  tens.  9x7  tens  are  63 
tens,  and  63  tens  +  2  tens  =  65  tens,  or  5  tens  and  6 
hundreds.  We  write  the  5  tens  as  the  tens  of  the  prod- 
uct, and  reserve  the  6  hundreds  in  the  mind,  to  be 
added  to  the  product  of  the  hundreds.  9x4  hundreds 
are  36  hundreds,  and  36  hundreds  +  6  hundreds  =  42 
hundreds,  or  2  hundreds  and  4  thousands,  and  these 
we  write  as  the  hundreds  and  thousands  of  the  product. 

This  is  the  method  generally  used. 

PM  OBLEMS. 

18.  A  druggist  sold  8  gallons  of  kerosene,  at  85  cents  a  gal- 
lon.    How  much  did  he  receive  for  it  ?  680  cents, 

19.  The  distance  from  New  York  to  Washington  is  226 
miles.  How  many  miles  does  a  man  travel,  who  goes  from 
New  York  to  Washington  and  back  ?  JiS2. 

20.  The  factors  are  9  and  147.     What  is  the  product  ? 

21.  How  many  gallons  in  6  hogsheads,  each  containing  124 
gallons  ?  7U' 

22.  A  market  gardener  bought  5  acres  of  land,  at  $635  per 
acre.     How  much  did  the  land  cost  him  ? 

23.  Nine  cars  of  a  freight  train  are  loaded  with  flour,  and 
each  car  contains  96  barrels.  How  many  barrels  of  flour  on 
the  train  ?  86 Jf. 

24.  How  many  pounds  in  6  barrels  of  Onondaga  salt,  each 
barrel  containing  256  pounds  ? 

25.  How  much  will  122  pairs  of  boots  cost,  at  $8  a  pair  ? 


52  INTEGERS. 

51*  In  the  picture  on  page  45  we  see  that  7  times  3 
blocks  are  the  same  as  3  times  7  blocks ;  or  that  the 
product  is  the  same,  whichever  of  the  two  numbers  is 
taken  for  the  multipUcand. 

In  problem  25,  $8  is  the  true  multiplicand,  because, 
if  one  pair  of  boots  costs  $8,  122  pairs  will  cost  122 
times  $8  ;  but  since  122  times  8  is  the  same  as  8  times 
122,  in  solving  the  problem  we  may  place  122  as  the 
multiplicand,  and  use  8  as  the  multipHer. 

52*    General  Principles  of  Multiplication. 

I.  In  the  solution  of  problems,  either  factor  may  he  used 
as  the  multiplicand. 

n.  The  true  multiplicand  is  that  factor  which  would  be 
used  in  solving  the  problem  by  Addition. 

m.  The  multiplicand  may  be  either  an  abstract  or  a 
concrete  number. 

rV.  The  multiplier  must  always  be  an  abstract  number. 

Y.  The  product  is  always  of  the  same  kind  as  the  true 

multiplicand.  (See  Manual,  page  216.) 

JPMOBIjJEMS. 

26.  Thomas  attended  public  school  76  days,  and  his  tuition 
was  3  cents  a  day.     How  much  was  his  school  bill  ?      228  cts. 

37.  In  one  week  a  n.ewsboy  sold  246  papers,  at  5  cents  each. 
How  much  did  he  receive  for  them  ? 

28.  A  man's  family  expenses  are  $4  a  day.  How  much  are 
they  for  a  year,  or  365  days  ?  $1,460. 

29.  How  much  will  2,755  army  blankets  cost,  at  $2  apiece  ? 

30.  How  many  pounds  in  6  bales  of  cotton,  each  weighing 
478  pounds?  2,868. 

31.  What  will  be  the  cost  of  building  a  horse  railroad  4 
miles  long,  at  $12,678  a  mile  ?  $50 J 12. 

32.  How  many  gallons  are  8  times  27,645  gallons  ? 

33.  How  many  pounds  are  5  times  32,051  pounds  ? 

34.  What  is  the  product  of  6  times  1,026,348  ?      6, 158,088, 


MULTIPLICATION.  53 

C^SE    II. 

The  Multiplier  any  number  of  Tens,  Hundreds,  Thou- 
sands, and  so  on. 

53.  Ex.  Multiply  254  by  10. 

Explanation.  —  We  write  the  numbers  as     soltition. 
shown  in  the   solution,   and    multiply   each      254 
figure  of  the  multiplicand  by  the  multiplier,  10 

as  in  Case  I.     10  x  4  ones  =  40  ones  ;  10  x  5      2540 
tens =50  tens,  and  50  tens +  4  tens  =  54  tens  ; 
10  X  2  hundreds  =  20  hundreds,  and  20  hundreds  +  5 
hundreds  =  25  hundreds. 

The  figures  of  the  product  are  the  same  as  those  of 
the  multiplicand,  with  a  cipher  on  the  right.     Hence, 

54*  Annexing  a  cipher  to  any  number  multiplies  it  by  10. 

55*  Annexing  a  second  cipher  multiplies  by  10  again ; 
that  is,  annexing  two  ciphers  to  any  number  multiplies  it 
by  10  times  10,  or  100. 

56t  Annexing  three  ciphers  to  a  number  multiplies  it  by 
10  times  100,  or  1,000. 

57.  Annexing  four  ciphers  multiplies  by  10,000 ;  annex- 
ing five  ciphers,  by  100,000  ;  and  annexing  six  ciphers,  by 
1,000,000. 

35.  How  much  will  10  barrels  of  mess  pork  cost,  at  |23  a 
barrel?  $230. 

36.  In  constructing  a  telegraph  line  100  miles  long,  how 
many  poles  will  be  required,  allowing  16  poles  to  the  mile  ? 

37.  How  many  panes  of  8  by  12  glass  in  1,000  boxes,  each 
box  containing  75  panes  ?  75, 000. 

38.  In  one  barrel  of  flour  there  are  196  pounds.  How  many 
pounds  in  10,000  barrels  ?  1, 960, 000. 

39.  Multiply  5,675  yards  by  100,000.        567,500,000  yards. 

40.  What  is  the  product  of  393  pounds  multiplied  by 
1,000,000  ? 


54 


INTEGERS. 


10 


SECOND  SOLUTION. 


58.  Ex.  Multiply  254  by  30. 

Explanation. — 30,  or  3  tens  =  3  times  ^^^^  solution. 

10  ones,  or  10  times  3  ones.     Hence,  30  ^^^ 

times  254  are  10  times  3  times  254.     We  - 

may  therefore  multiply  by  3,  as  in  Case  ^^^^ 
I.,  and  the  product  thus  obtained  by  10. 

The  final  result,  7,620,  is  10  times  3  times  7620 
254,  or  30  times  254. 

In  the  second  solution,  after  multiply-  254 

ing  254  by  3,  we  have  annexed  a  cipher  to  30 

the  result.  rjQ^Q 

59.  To  multiply  by  300,  we  multiply  by  3,  and  annex 
two  ciphers  to  the  product ;  to  multiply  by  3,000,  we  mul- 
tiply by  3,  and  annex  three  ciphers,  and  so  on. 

PUOBI.EMS, 

41.  At  $65  a  hogshead,  how  much  will  300  hogsheads  of 
molasses  cost  ?  $19,500. 

43.  How  many  bushels  of  oats-  will  be  required  to  keep  800 
cavalry  horses  a  year,  if  187  bushels  are  required  for  each 
horse  ? 

43.  How  many  sheets  of  paper  in  3,000  copies  of  Worcester's 
Dictionary,  there  being  113  sheets  in  each  copy  ?      336,000. 

44.  A  man  bought  a  farm  of  70  acres,  at  $135  an  acre. 
How  much  did  the  farm  cost  him  ?  $8, 750. 

45.  How  many  pounds  of  beef  in  814  barrels,  each  barrel 
containing  300  pounds  ?  J^2,800. 

46.  A  railroad  company  bought  9,000  cords  of  wood,  at  $5 
a  cord.     How  much  did  the  wood  cost  them  ? 

47.  A  vessel  at  New  York  took  on  a  cargo  of  7,000  barrels 
of  kerosene.  If  each  barrel  contained  43  gallons,  how  many 
gallons  of  kerosene  in  the  cargo  ?  301, 000. 

48.  A  wholesale  grocer  bought  500  chests  of  tea,  each  con- 
taining 56  pounds.  How  many  pounds  did  all  the  chests  con- 
tain ?  28,000. 


MULTIPLICATION 


55 


49.  In  one  hour  there  are  60  minutes.  How  many  minutes 
in  1  day,  or  24  hours  ?  ljJi40. 

50.  How  many  pounds  of  cotton  in  40,000  bales,  each 
weighing  394  pounds  ?  15, 760, 000. 

51.  An  Illinois  farmer  had  80  acres  of  com,  which  yielded 
94  bushels  an  acre.  How  many  bushels  had  he  in  the  whole 
crop?  7,520. 

53.  Multiply  349  by  4,000,000.  996,000,000. 

53.  What  is  the  product  of  600,000  times  972  ? 

54.  The  factors  are  90,000  and  3,165.    What  is  the  product  ? 


C^SE    III. 
The  Multiplier  more  than  One  Figure. 

FIRST     METHOD. 


Multiply  563  by  34. 


FIB8T     SOLUTION. 


Multiplying 
ly  4. 

563 
4 


2252 


Multiplying 
by  30. 

563 
30 


16890 


Adding 
partial  products. 

2252 
16890 


19142 


60.  Ex. 

Explanation. — ^As 
34  consists  of  4  ones 
and  3  tens,  or  of  4 
and  30  ;  and  as  we 
can  not  multiply  563 
by  the  whole  34  at 

once,  we  first  multiply  it  by  4  and  then  by  30,  and  after- 
ward add  the  two  results  or  partial  products.  19,142, 
the  result  thus  obtained,  is  the  sum  of  4  x  563  plus 
30  X  563,  or  34  x  563. 

In  the  First  Solution,  each  of  the  second  solittion. 
three  steps  stands  by  itself;  in  the 
Second  Solution  they  are  placed  to- 
gether. 

In  multiplying  by  3  tens,  or  30,  we 
may  first  write  a  cipher  in  ones'  place, 
and  then  to  the  left  of  it  write  the 
product  obtained  by  multiplying  by  3. 
solve  the  following  problems. 


563 
34 

2252]     Partial 

16890 )  P^«<i^<^t«- 
19142 


In  this  manner 


56  INTEGERS. 

JPJtOBUJEMS. 

55.  How  many  yards  in  24  pieces  of  sheeting,  eacli  contain- 
ing 43  yards?  1,008. 

56.  How  many  pounds  in  a  load  of  74  bushels  of  oats, 
allowing  32  pounds  to  a  bushel  ?  2,368. 

57.  How  much  are  132  ounces  of  gold  worth,  at  $16  an 
ounce  ? 

58.  How  many  hills  of  com  in  a  field  that    contains   36 
rows,  and  185  hills  in  each  row  ? 

59.  A  wholesale  dealer  in  watches  sold  48  watches,  at  $125 
each.     How  much  did  he  receive  for  them  ?  $6, 000. 

60.  Twelve  things  make  a  dozen.    How  many  eggs  in  a 
barrel  containing  87  dozens  ?  1,0J!^J^. 

61.  A  western  township  contains  36  square  miles.     How 
many  square  miles  in  a  county  of  25  townships  ?  900. 

62.  How  many  ounces  of  silver  can  be  obtained  from  392 
tons  of  ore,  at  the  rate  of  34  ounces  per  ton. 

SECOND    METHOD. 

61.  Ex.  What  is  the  product  of  243  times  2,156? 

Explanation. — ^In  the  First  Solution,  the  ^^^^^  soltttion. 

ciphers  on  the  right  of  the  second  and  e^ 

third  partial  products  serve  merely  to  fill  

the  places  of  ones  and  tens  ;  and  since  Q^QS 

the  sum  of  any  number  of  O's  is  0,  we  4.01  nnn 

really  omit  these  ciphers  in  adding  the  

partial  products.  523908 

We  may  omit  to  write  these  ciphers  in  ^  ^„ 

^  ^  SECOND  SOLlfTION. 

the  partial  products,  as  shown  m  the  Sec-  2156 

ond  Solution,  without  affecting  the  total  243 

product.      By  this  method,   the  second  ^T^ 

partial  product  may  be  found  by  multi-  8624 

plying  by  4  instead  of  40  ;  and  the  third,  4312 

by  multiplying  by  2  instead  of  200.    But  503908 
we  must  always 


MULTIPLICATION. 


67 


62.  Write  the  first  figure  of  each  partial  product  di- 
rectly under  the  figure  of  the  multiplier  used  to  obtain  it. 

This  is  the  method  in  general  use. 

In  reading  each  partial  product,  its  true  value  should 
be  given.  This  is  done  by  reading  it  as  though  the 
ciphers  were  written. 

JPJROJSZJEMS. 

63.  A  drover  bought  13  young  cattle,  at  $34  a  head.  How 
much  did  they  cost  him  ?  $44^. 

64.  An  iron-founder  bought  37  tons  of  pig-iron,  at  $45  a 
ton.    How  much  did  he  pay  for  it  ?  $1,665. 

65.  A  merchant  bought  26  pieces  of  calico,  each  containing 
39  yards.    How  many  yards  in  all  ?  1,01J^. 

66.  How  many  miles  will  a  railroad  train  run  in  48  hours, 
running  at  the  rate  of  34  miles  an  hour  ?  1,632. 

67.  How  much  must  I  pay  for  a  farm  of  67  acres,  at  $75  an 
acre  ? 

68.  A  tobacco  grower  raised  10  acres  of  tobacco,  which 
yielded  1,654  pounds  to  the  acre.  How  much  did  his  tobacco 
crop  weigh  ?  16, 5^0  pounds. 

69.  What  is  the  product  of  58  tunes  157  ?  9, 106. 

70.  73  X  593  =  how  many  ?  JiS,289. 

71.  84  X  748  square  miles  =  how  many  square  miles  ? 
(72)  (73)  (74)  (75)  (76) 


4,306 

284 


2,087 
659 


,009 
573 


63.  Ex.  Multiply  6,U9  by 
503. 

Explanation. — ^In  the  Sec- 
ond Solution  we  have  mul- 
tiplied first  by  the  3  (ones), 
and  then  by  the  5  (hun- 
dreds), omitting  to  multiply 


10,609 
138 

FIRST  SOLUTION. 


50,028 
971 


6149 
503 

18447 
0000 
30745 

3092947 


SECOND  SOLUTION. 

6149 
503 


18447 
30745 

3092947 


68  INTEGERS. 

by  the  0  (tens),  because  0  times  6,149  is  0,  as  shown  in 
the  First  Solution. 

64t  Since  0  times  any  number  is  0,  we  may  always  omit  to 
multiply  by  ciphers  that  stand  between  other  figures  in  the 
multiplier,  being  careful  to  write  the  first  figure  of  each  par- 
tial product  directly  under  the  figure  of  the  multiplier  used 
to  obtain  it. 

PMOBIjEMS, 

77.  At  the  close  of  the  war,  in  1865,  a  quarter-master  sold 
905  mules,  at  $183  apiece.  How  much  was  received  for 
them?  $165,615. 

78.  In  1864,  a  certain  county  raised  706  men  for  the  army, 
paying  a  bounty  of  $855  to  each  man.  How  much  bounty 
money  was  paid  by  the  county  ?  $608,630. 

79.  At  the  office  of  a  daily  newspaper,  217  reams  of  paper 
are  used  each  day.  How  many  reams  will  be  required  to  last 
308  days  ? 

80.  What  will  be  the  weight  of  the  wire  for  a  line  of  tele- 
graph 207  miles  long,  if  one  mile  of  wire  weighs  326  pounds  ? 

67,482  pounds. 

81.  In  building  a  factory  a  mason  used  409  loads  of  brick, 
each  load  containing  648  bricks.  How  many  bricks  did  he 
use?  265,032. 

82.  The  daily  wages  of  the  hands  in  a  cotton -factory 
amount  to  $736.  How  much  is  that  for  a  year  of  307  work- 
ing-days ? 

83.  Multiply  17,248  by  6,003.  103, 539, 7 U. 

84.  Find  the  product  of  53,276  multiplied  by  5,002. 

65.  Ex.  What  is  the  product  of  4,300  times  2,394? 

Explanation. — To  multiply  by  300,  we  solution. 

first  multiply  by   3,  and  then   annex  two  2394 
ciphers  ;  (Case  II).   In  the  same  manner,  to  ^^^^ 

multiply  by  4,300  we  first  multiply  by  43,  7182 

and  then  annex  two  ciphers,  as  shown  in  ^^76 


the  solution.  10294200 


MULTIPLICATION.  59 

66«  ^ule  for  MuUipticatio7i  of  Integers. 

I.  Write  the  greater  number  for  the  multiplicand,  and 
under  it  the  less  number  for  the  multiplier. 

n.  Wlien  the  multiplier  consists  of  only  one  figure, 
Commence  with  the  ones,  and  multiply  successively  each 
figure  of  the  multiplicand  by  the  multiplier.  Write  in  the 
product  the  right-hand  figure  of  each  result,  and  add  the 
left-hand  figure  to  the  next  result,  as  in  Addition. 

m.  When  the  multiplier  consists  of  more  than  one 
figure, 

1.  Commence  at  the  right,  multiply  the  multiplicand  by 
each  figure  of  the  multiplier,  except  ciphers,  and  place  the 
right-hand  figure  of  each  partial  product  under  that  figure 
of  the  midtiplier  used  to  obtain  it. 

2.  Add  the  partial  products. 

IV.  "When  the  multiplier  ends  with  one  or  more 
ciphers, 

After  multiplying  by  the  other  figures,  annex  an  equal 
number  of  ciphers  to  the  product. 

JPMOBZEMS. 

85.  A  pork  packer  bought  2,400  sacks  of  Turks'  Island 
salt,  each  sack  containing  168  pounds.  How  many  pounds 
of  salt  did  he  buy  ?  403,200. 

86.  A  railroad  company  bought  5,800  bars  of  railroad 
iron,  each  bar  -weighing  403  pounds.  How  much  did  the 
whole  weigh  ?  2, 331, 600  pounds. 

87.  A  whaler  entered  New  Bedford  with  a  cargo  of  1,760 
barrels  of  whale  oil,  each  barrel  containing  39  gallons.  How 
many  gallons  of  oil  in  the  cargo  ?  68, 6^0. 

88.  How  much  would  the  yearly  pay  of  an  army  of  650,000 
men  amount  to,  at  $192  per  man  ?  $124,800,000. 

89.  If  the  factors  are  9,654  and  21,800,  what  is  the  pro- 
duct? 

90.  How  many  cords  of  wood  can  be  cut  from  85  acres  of 
wood-land,  at  the  rate  of  65  cords  to  the  acre  ? 


60  INTEGERS. 

91.  How  many  barrels  will  be  made  at  a  barrel  factory  in 
56  clays,  if  159  barrels  are  made  each  day  ?  8,904. 

92.  How  much  will  94  passenger  cars  cost,  at  $3,475  each  ? 

93.  If  97  tons  of  railroad  iron  are  required  for  one  mile  of 
track,  how  many  tons  will  be  required  for  a  road  359  miles 
long?  34,8S3. 

94.  An  agent  in  one  year  sold  795  sewing  machines,  at  $65 
apiece.     How  much  were  his  receipts  ?  ^gi  675_ 

95.  The  number  of  trips  made  by  the  boats  at  a  certain 
ferry  is  234  per  day,  and  the  average  number  of  passengers 
per  trip  is  108.  How  many  persons  cross  the  ferry  in  a 
day  ?  ^g  ^1^2 

96.  Multiply  1,327  by  246.  •  S26,4J^. 

97.  What  is  the  product  of  the  factors  2,076  and  382  ? 


5800 


156600 


67.  Ex.  The  factors  are  5,800  and  27.  ^^^^t  solution. 
What  is  the  product  ?  27 

Explanation.— In  the  First  Solution  we 

have  multiplied  27  by  5,800,  by  first  mul-  ^3^ 
tiplying  by  58,  and  to  the  product  annex- 
ing two  ciphers.     But  as  58  times  27  = 
27  times  58,    (Prin.  I.),   in  the   Second  second  solxttion. 

Solution  we  have  multipHed  58  by  27,  5800 

and  to  the  product  annexed  two  ciphers.  27 

The  results  in    the    two   solutions    are  406 

alike,   and  either  method  may  be  em-  116 

ployed.  156600 

PROBJLEMS. 

98.  In  the  year  1865  there  had  been  16  Presidents,  whose 
united  terms  of  oflSce  amounted  to  76  years.  How  much  had 
their  salaries  amounted  to,  at  $25,000  a  year  ?     $1,900,000. 

99.  At  the  rate  of  1,400  words  an  hour,  how  many  words 
can  be  sent  over  a  telegraph  line  in  24  hours  ?  33,600. 

100.  How  much  will  a  clergyman's  salary  amount  to  in  17 
years,  at  $1,200  a  year  ?  &20,JfiO. 


MULTIPLICATION.  61 

101.  There  are  5,280  feet  in  a  mile.  How  many  feet  long  is 
the  Pennsylvania  Railroad  track,  which  is  353  miles  long  ? 

102.  The  multiplicand  is  39,200,  and  the  multiplier  is  65. 
What  is  the  product  ?  2, 51^8, 000. 

103.  Multiply  58,000  by  73.  4,231^,000. 

104.  80,004  X  12,357  =  how  many?  988,609,428. 

105.  At  a  saw-mill  185  logs  were  sawed  into  boards,  each 
log  making  642  feet  of  lumber.  How  many  feet  did  all  of 
them  make  ? 

106.  If  1,594  pounds  of  saltpeter  are  used  in  making  1  ton 
of  guni)owder,  how  many  pounds  will  be  used  in  making 
1,056  tons  ? 

68.  Ex.  Multiply  67,000  by  2,100. 

Explanation. — ^We  first  multiply  67,000 
by  21,  as  before  explained,  and  to  1,407,000, 
the  product  tbus  obtained,  we  annex  two 
ciphers  ;  or  to  the  product  of  21  x  67  we 
annex  three  ciphers  for  those  at  the  right 
of  the  67  (thousands)  in  the  multiplicand, 
and  two  for  those  at  the  right  of  the  21  140700000 
(hundreds)  in  the  multipher.  (See  Manual,  page  217.) 

JPMOJiZJEMS. 

107.  One  year  a  piano-forte  manufacturer  made  360  pianos, 
at  a  cost  of  $270  each.     How  much  did  they  all  cost  him  ? 

$97,200. 

108.  How  much  will  it  cost  to  build  80  miles  of  turnpike 
road,  at  $1,500  a  mile  ?  $120,000. 

109.  If  a  rope-maker  can  spin  3,700  feet  of  rope  in  a  day, 
how  many  feet  can  he  spin  in  260  days  ?  962,000. 

110.  A  barge  was  loaded  with  600  bales  of  hay,  which 
weighed  290  pounds  each.  What  was  the  whole  weight  of 
the  hay?  174,000  pmcnds, 

111.  What  is  the  product  of  63,000  times  1,850  ? 

116,550,000. 


62  INTEGKRS'. 

112.  A  manufacturer  of  reapers  and  mowers  sold  in  one 
year  2,476  machines,  at  $135  each.  How  much  did  he  receive 
for  them?  $309,500. 

113.  How  many  poles  will  be  required  for  a  hop-yard  of  40 
acres,  allowins-  1,280  poles  to  the  acre  ? 

114.  A  common  clock  strikes  150  times  every  day.  How 
many  times  does  it  strike  in  one  year,  or  365  days  ? 

115.  Tlie  three  factors  of  a  number  are  39,  24,  and  17. 
What  is  the  number  ?  15, 912. 

69.  "When  a  final  product  is  obtained  by  the  use  of 
more  than  two  factors,  the  process  is  called  Continued 

Multiplication.      (Sec  Manual,  page  217.) 

116.  A  grocer  packed  28  barrels  of  eggs,  putting  83  dozen 
in  each  barrel.      How  many  eggs  did  he  pack  ?         22, 908. 

117.  How  many  sheets  in  45  reams  of  letter  paper,  there 
being  20  quires  in  each  ream,  and  24  sheets  in  each  quire  ? 

118.  How  many  miles  will  a  conductor  on  a  railroad  81 
miles  long  travel  in  13  years,  if  he  makes  a  round  trip  every 
day  for  300  days  each  year  ?    (See  Manual,  page  23  7.)   63 1, 800  miles. 

119.  In  7  dozen  papers  of  pins,  how  may  pins,  each  paper 
containing  10  rows,  and  each  row  36  pins  ?  30,2^0. 

120.  What  number  is  that  of  which  the  factors  are  5,006, 
79,  and  840  ? 

JRJEVIEW    PJtOBIjEMS. 

(See  Manual,  page  217.) 

1.  A  mechanic  earns  |45  a  month,  and  his  expenses  are 
$476  a  year.  How  much  can  he  save  in  one  year,  or  12 
months  ?  $6Jf. 

2.  A  stock  train  consists  of  19  cars,  each  containing  17  cat- 
tle. How  much  do  aU  the  cattle  weigh,  their  average  weight 
being  895  pounds  ? 

3.  A  man  bought  a  house  and  lot,  for  which  he  paid  $1,275 
down,  and  $250  each  year  for  7  years.  How  much  did  the 
place  cost  him  ?  $3,025. 


ICATIONf  ]jy  ^ 
rs  old  he  ha^$[WOf 


cm  z 

MULTIPLICATIONfTT'RrT^]5f63 


4.  When  Victor  was  15  years  old  he  ha<?$[fOOf  He  ^red 
$37  each  year  until  he  was  31.  How  much  money  had  he 
then  ?  $322. 

5.  A  book-keeper  earns  $150  a  month,  and  expends  $68. 
How  much  does  he  save  in  a  year  ?  $984. 

6.  An  agent  sold  38  sets  of  the  New  American  Cyclopaedia, 
each  set  containing  16  volumes,  at  $5  a  volume.  How  much 
did  he  receive  for  them  ? 

7.  A  steamer  was  burned  at  sea,  having  on  board  296  pas- 
sengers and  a  crew  of  73  persons.  Of  these,  117  passengers 
and  49  of  the  crew  were  saved.  How  many  persons  were 
lost  ?  203. 

8.  A  hotel  keeper  charges  $5  a  week  for  board.  How  much 
will  he  receive  from  27  boarders  in  one  year,  or  52  weeks  ? 

9.  A  merchant  tailor  bought  84  pieces  of  broadcloth,  each 
piece  containing  37  yards,  at  $6  a  yard.  How  much  did  the 
cloth  cost  him  ? 

10.  If  33  pickets  are  used  in  building  one  rod  of  fence,  how 
many  pickets  will  be  required  for  the  fence  around  a  lot  16 
rods  long  and  15  rods  wide  ?  2,0^6. 

11.  How  much  will  a  baker  receive  for  48  pounds  of  crack- 
ers, at  14  cents  a  pound,  and  128  loaves  of  bread,  at  9  cents  a 
loaf? 

12.  A  provision  dealer  bought  850  barrels  of  pork,  at  $17 
a  barrel,  and  sold  the  lot  for  $696  more  than  cost.  How 
much  did  he  receive  ? 

13.  A  produce  dealer  bought  4,124  bushels  of  wheat,  2,476 
bushels  of  com,  and  1,218  bushels  of  oats.  After  selling  3,684 
bushels  of  wheat,  1,787  bushels  of  com,  and  975  bushels  of 
oats,  how  many  bushels  of  each  had  he  left  ?       689  Im.  of  com. 

14.  A  is  wortli  $675 ;  B  is  worth  twice  as  much ;  and  C  is 
worth  4  times  as  much  as  both.  How  much  are  B  and  C  each 
worth  ?   How  much  are  the  three  worth  ?  Worth  of  all^  $10, 125. 

15.  A  wholesale  grocer,  having  156  chests  of  tea,  each  con- 
taining 84  pounds,  sold  57  chests  to  one  customer,  and  23 
chests  to  another.  How  many  chests,  and  how  many  pounds 
of  tea  had  he  left  ?  6, 384  pounds. 


64 


INTEGERS 


SECTION  7. 
^irisiojv, 

INDXJCTION. 

(See  Manual,  page  217.) 

70.  1.  Some  lumbermen  made  a  raft  of  20  logs,  putting  5 
logs  in  each  crib.     Of  how  many  cribs  did  the  raft  consist  ? 

2.  Of  how  many  cribs  will  the  raft  consist  if  they  use  25 
logs? 

3.  How  many  cribs  will  there  be  in  the  raft  if  they  use  30 
logs  ? 

4.  To  how  many  girls  can  I  give  24  peaches,  if  I  give  4 
peaches  to  each  girl  ? 

5.  One  afternoon  Robert  found  2l  chestnuts  in  Ipurs,  each 
bur  containing  3  chestnuts.     How  many  burs  did  he  find  ? 

6.  In  a  raft  of  4  cribs  there  are  20  logs.     Of  how  many 
logs  does  each  crib  consist  ? 

7.  In  a  raft  of  30  logs  there  are  6  cribs.    How  many  logs  in 
each  crib  ? 

8.  A  furniture  dealer  sold  4  rocking-chairs  for  $28.    How 
many  dollars  did  he  receive  for  each  of  them  ? 


DIVISION.  66 

9.  A  military  company  of  42  men  marched  in  6  equal  files. 
How  many  men  were  in  each  file  ? 

10.  A  grocer  paid  $33  for  8  barrels  of  apples.  What  was 
the  price  per  barrel  ? 

In  solving  each  of  the  first  five  examples,  we  find 
how  many  times  one  of  two  numbers  is  contained  in 
the  other  ;  and  in  solving  each  of  the  other  examples, 
we  separate  one  of  two  numbers  into  as  many  equal 
parts  as  there  are  ones  in  the  other. 

71.  The  process  of  finding  how  many  times  one  of 
two  numbers  is  contained  in  the  other,  or  of  finding 
one  of  the  equal  parts  into  which  a  number  may  be 
separated,  is  Division. 

72.  The  result  thus  found  is  the  Quotient. 

73.  The  number  to  be  divided  is  the  Dividend  ;  and 

74.  The  number  by  which  the  dividend  is  to  be 
divided,  is  the  Divisor. 

The  number  of  ones  in  the  dividend  must  be  equal 
to  the  number  of  ones  in  the  divisor  taken  as  many 
times  as  there  are  ones  in  the  quotient. 

11.  How  many  times  8  loaves  of  bread  are  56  loaves  ? 

12.  What  is  the  quotient  of  63  divided  by  7  ? 

13.  The  dividend  is  54,  and  the  divisor  is  6.  What  is  the 
quotient  ? 

75.  When  any  number  of  things  is  divided  into  2 
equal  parts,  one  of  those  parts  is  one  half  of  that  num- 
ber of  things.  One  of  the  3  equal  parts  into  which  any 
number  may  be  divided  is  one  third  of  that  number ; 
one  of  the  4  equal  parts  is  one  fourth ;  one  of  the  5 
equal  parts  is  one  fifth,  and  so  on.     (See  Manual,  page  217.) 

76.  One  half  of  a  number  is  found  by  dividing  it  by 
2  ;  one  third  by  dividing  it  by  3  ;  one  fourth  by  divid- 
ing it  by  4  ;  one  fifth  by  dividing  it  by  5  ;  one  sixth  by 
dividing  it  by  6,  and  so  on. 

F 


INTEGERS. 


14.  A  man's  farm  contains  70  acres,  and  one  seventh  of  it  is 
woodland.     How  many  acres  of  woodland  has  he  ? 

15.  If  9  table  spreads  cost  $36,  what  part  of  $36  will  1 
table  spread  cost  ?     How  many  dollars  will  one  cost  ? 

16.  How  will  you  find  one  eighth  of  48  ?  In  this  question, 
what  is  the  dividend  ?     What  is  the  divisor  ? 

17.  If  30  yards  of  carpeting  be  cut  into  5  breadths,  how 
many  yards  will  there  be  in  each  breadth  ? 

77.  This  sign  -~,  written  between  two  numbers,  sig- 
nifies that  the  number  before  it  is  to  be  divided  by  the 
number  after  it. 

It  is  called  the  Sign  of  Division,  and  is  read  divided 
by.  Thus,  72  -^  6  =:  12  is  read,  72  divided  by  6 
equals  12. 

18.  Read  99  -^  9  =  11.  |  19.  Read  64  -j-  8  =  8. 

20.  81  -=-  9  =  how  many  ? 

21.  98  -T-  7  =  14.  In  this  expression,  which  number  is  the 
dividend,  which  the  divisor,  and  which  the  quotient  ? 

22.  A  teamster  feeds  his  horses  6  bushels  of  oats  each  week. 
How  many  weeks  will  60  bushels  last  them  ? 

23.  If  a  teamster  feeds  his  horses  60  bushels  of  oats  in  10 
weeks,  how  many  bushels  does  he  feed  in  1  week  ? 

(See  Manual,  page  21T.) 
78.     DIVISION     TABLE. 


0123456789 [1 

0    6  12  18  2i  30  36  42  48  54  I  6 

0133456789 

0123456789 

0    2    4    6    8  10  12  14  16  18  [  2 

0    7  14  21  28  35  42  49  56  63  (^  7 

0123456789 

0123456789 

0    3    6    9  12  15  18  21  24  27  1  3 

0    8  16  24  32  40  48  56  64  72  13 

0123456789 

0123456789 

0    4    8  12  16  20  24  28  32  36  1  4 

0    9  18  27  36  45  54  63  72  81  [^  9 

0123456789 

0123456789 

0    5  10  15  20  25  30  35  40  45  1  5 
0123456789 

0  10  20  30  40  50  60  70  80  90  [IQ 
012    3    45    6    789 

DIVISION.  67 


OBAJL    EXEJRCISES. 

1. — ^1.  Divide  by  2  from  3  in  0  to  2  in  20,  thus :  2  in  0,  0  times ;  2  in 

2,  once ;  2  in  4,  2  times,  and  so  on. 

2.  Divide  by  2  from  2  in  20  to  2  in  0,  thus :  2  in  20, 10  times ;  2  in 
18,  9  times ;  2  in  16,  8  times,  and  so  on. 

3.  Divide  by  2  from  1  half  of  0  to  1  half  of  20,  thus :  1  half  of  0  is  0 ; 
1  half  of  2  is  1 ;  1  half  of  4  is  2,  and  so  on. 

4.  Divide  by  2  from  1  half  of  20  to  1  half  of  0,  thus :  1  half  of  20  is 
10 ;  1  half  of  18  is  9 ;  1  half  of  16  is  8,  and  so  on. 

2. — 1.  Divide  by  3  from  3  in  0  to  3  in  30,  thus ;  3  in  0,  0  times ;  3  in 

3,  once ;  3  in  6,  2  times,  and  so  on. 

2.  Divide  by  3  from  3  in  30  to  3  in  0,  thus ;  3  in  30, 10  times ;  3  in 
27,  9  times ;  3  in  24,  8  times,  and  so  on. 

3.  Divide  by  3  from  1  third  of  0  to  1  third  of  30,  thus ;  1  third  of  0 
is  0 ;  1  third  of  3  is  1 ;  1  third  of  6  is  2,  and  so  on. 

4.  Divide  by  3  from  1  third  of  30  to  1  third  of  0,  thus :  1  third  of  30 
is  10;  1  third  of  27  is  9;  1  third  of  24  is  8,  and  so  on. 

3. — 1.  Divide  by  4  from  4  in  0  to  4  in  40,  thus :  4  in  0,  0  times ;  4  in 

4,  once ;  4  in  8,  2  times,  and  so  on. 

2.  Divide  by  4  from  4  in  40  to  4  in  0,  thus :  4  in  40, 10  times ;  4  in 
36,  9  times  ;  4  in  32,  8  times,  and  so  on. 

3.  Divide  by  4  from  1  fourth  of  0  to  1  fourth  of  40,  thus :  1  fourth 
of  0  is  0;  1  fourth  of  4  is  1 ;  1  fourth  of  8  is  2,  and  so  on. 

4.  Divide  by  4  from  1  fourth  of  40  to  1  fourth  of  0,  thus :  1  fourth 
of  40  is  10 ;  1  fourth  of  36  is  9 ;  1  fourth  of  32  is  8,  and  so  on. 

4. — 1.  Divide  by  5  from  5  in  0  to  5  in  50.          (See  Manual,  page  21T.) 

2.  Divide  by  5  from  5  in  50  to  5  in  0. 

3.  Divide  by  5  from  1  fifth  of  0  to  1  fifth  of  50. 

4.  Divide  by  5  from  1  fifth  of  50  to  1  fifth  of  0. 

5. — 1.  Divide  by  6  from  6  in  0  to  6  in  60. 

2.  Divide  by  6  from  6  in  60  to  6  in  0. 

3.  Divide  by  6  from  1  sixth  of  0  to  1  sixth  of  60. 
4  Divide  by  6  from  1  sixth  of  60  to  1  sixth  of  0. 

6. — 1.  Divide  by  7  from  7  in  0  to  7  in  70. 

2.  Divide  by  7  from  7  in  70  to  7  in  0. 

3.  Divide  by  7  from  1  seventh  of  0  to  1  seventh  of  70. 
4-  Divide  by  7  from  1  seventh  of  70  to  1  seventh  of  0. 

7. — 1.  Divide  by  8  from  8  in  0  to  8  in  80. 

2.  Divide  by  8  from  8  in  80  to  8  in  0. 

3.  Divide  by  8  from  1  eighth  of  0  to  1  eighth  of  80. 

4.  Divide  by  8  from  1  eighth  of  80  to  1  eighth  of  0. 


68  INTEGERS. 

8. — 1.  Divide  by  9  from  9  in  0  to  9  in  90. 

2.  Divide  by  9  from  9  in  90  to  9  in  0. 

3.  Divide  by  9  from  1  ninth  of  0  to  1  ninth  of  90. 

4.  Divide  by  9  from  1  ninth  of  90  to  1  ninth  of  0. 

9. — 1.  Divide  by  10  from  10  in  0  to  10  in  100. 

2.  Divide  by  10  from  10  in  100  to  10  in  0. 

3.  Divide  by  10  from  1  tenth  of  0  to  1  tenth  of  100. 

4.  Divide  by  10  from  1  tenth  of  100  to  1  tenth  of  0. 

o-a.se   I. 
The  Divisor  One  Figure. 

FIEST   METHOD. 

79.  Ex.  What  is  the  quotient  of  696  divided  by  3. 

Explanation. — Since  the  dividend  is  too 
large  to  be  divided  mentally,  we  place  the 
divisor  at  the  right  of  it,  separating  them 
by  a  line,  and  draw  a  line  under  the  divi- 
sor to  separate  it  from  the  quotient.  Then 
commencing  at  the  left  hand,  we  divide 
each  figure  of  the  dividend  by  the  divisor,  _6 
thus  ;  3  is  contained  in  6  hundreds,  2 
hundred  times.  We  write  this  result  below  the  di- 
visor, for  the  first  partial  quotient.  We  have  now  used 
600  (or  3  times  200)  of  the  dividend ;  and,  subtract- 
ing 600,  we  have  a  remainder  of  96,  or  9  tens  and 
6  ones,  yet  to  be  divided.  3  is  contained  in  9  tens  or 
90,  3  tens  or  30  times.  We  write  the  3  tens  or  30  for 
the  second  partial  quotient.  We  have  now  used  90 
(90  tens),  or  3  times  30  ;  and,  subtracting  it  from  96, 
the  partial  dividend,  we  have  6  ones  for  another  par- 
tial dividend.  3  is  contained  in  6  ones,  2  times.  We 
write  the  2  (ones)  as  the  third  partial  quotient,  and 
subtract  the  6  ones  from  the  last  partial  dividend.  We 
have  now  used  aU  the  figures  of  the  dividend,  and  we 
have  therefore  found  all  the  partial  quotients.      The 


EST  SOLUTION. 

696 

3 

600 

200 

96 

30 

90 

2 

6 

232 

DIVISION.  69 

sum  of  these  partial  quotients,   or  232,   is  the  total 
quotient  required. 

Explanation.— We   first  divide   the   6  second  solution. 

(hundreds)  of  the  dividend  by  3,  as  be-       "^"  I  5 

fore,  and  the  quotient,  2  (hundreds),  we       i 1 232 

-write  as  the  first  figure  of  the  final  quo-        9 

tient.     3  times  2  (hundreds)  are  6  (hun-        ^ 

dreds),  and  since  this  is  to  be  subtracted  6 

from  the  dividend,  we  write  it  under  the  _6_ 

hundreds'  figure  of  the  dividend.  6  (hun- 
dreds) from  6  (hundreds)  leaves  0.  The  next  part  of 
the  dividend  to  be  divided  is  the  9  (tens),  which  we 
bring  down  for  a  partial  dividend.  3  is  contained  in 
9  (tens),  3  (tens)  times,  and  we  write  this  3  as  the  second 
figure  of  the  final  quotient.  3  times  3  (tens)  are  9  (tens), 
and  since  this  is  to  be  subtracted  from  the  tens  of 
the  dividend,  we  write  it  under  the  partial  dividend. 
9  (tens)  from  9  (tens)  leaves  0.  The  next  part  of  the 
dividend  to  be  divided  is  the  6  (ones),  which  we  bring 
down  for  the  last  partial  dividend.  3  is  contained  in  6 
(ones),  2  (ones)  times,  and  we  write  this  2  as  the  third 
figure  of  the  final  quotient.  3  times  2  (ones)  are  6 
(ones),  and  since  this  is  to  be  subtracted  from  the  ones 
of  the  dividend,  we  write  it  under  the  last  partial  divi- 
dend. 6  (ones)  from  6  (ones)  leaves  0.  We  have  now 
used  all  the  figures  of  the  dividend,  and  the  result,  232, 
is  the  quotient  required. 

In  this  solution  we  have  taken  the  same  steps  as  in 
the  first,  but  we  have  omitted  to  write  the  ciphers  of 
the  partial  quotients,  because  the  value  of  each  figure 
is  known  without  them,  and  we  thus  use  fewer  figures. 
By  writing  the  2  (hundreds)  in  hundreds'  place,  the  3 
(tens)  in  tens'  place,  and  the  2  (ones)  in  ones'  place,  we 
really  add  200,  30,  and  2. 


70 


INTEGERS. 


PIiOSJ0:EMS. 

1.  At  $4  apiece,  how  many  hats  can  be  bought  for  $848  ? 

2.  K  3  workmen  can  lay  3,396  bricks  in  a  day,  what  part 
of  3,396  bricks  can  one  man  lay  ?  How  many  bricks  can  one 
man  lay?  1,132. 

3.  Four  newsboys  bought  84  newspapers,  each  taking  an 
equal  share  of  them.     How  many  papers  did  each  boy  take  ? 

4.  If  2  men  can  chop  28  cords  of  wood  in  a  week,  how 
many  cords  can  one  man  chop  ? 

5.  A  door  maker  received  $639  for  doors,  at  $3  each.  How 
many  doors  did  he  sell  ?  213. 

6.  At  $4  a  cord,  how  many  cords  of  wood 
can  be  be  bought  for  $1,648  ? 

7.  How  many  barrels  of  flour  must  a  miller 
sell  each  day  to  sell  216  barrels  in  3  days  ? 

8.  If  a  house  painter  earns  $255  in  5  months, 
how  much  does  he  earn  in  one  month  ?       $51. 

9.  If  4  tons  of  coal  are  used  each  day  in  an 
iron-foundery,  how  many  days  will  3,284  tons  last  ? 


SOLUTION 
or  PROBLEM 


1648 
16 

4 
4 

8 
8 


4__ 
412 


SOLUTION. 

917 

7 


21 
21 


7 
131 


80.  Ex.  Divide  917  by  7. 

Explanation. — Since  9  is  greater  than 
7,  7  is  contained  in  9  at  least  1  time ;  and 
since  2  times  7  are  14,  and  14  is  greater 
than  9, 7  is  not  contained  in  9, 2  times.  "We 

therefore  write  the  1  as  the  first  figure  of        

the  quotient.     7  times  1  (hundred)  are  7  7 

(hundreds)  ;  and  since  this  is  to  be  sub-  _7_ 

tracted  from  the  dividend,  we  write  it 
under  the  hundreds'  figure  of  the  dividend.  7  (hun- 
dreds) from  9  (hundreds)  leave  2  (hundreds).  We 
next  bring  down  the  1  (ten)  at  the  right  of  the  2  (hun- 
dreds), and  we  have  21  (tens)  for  a  partial  dividend. 
7  is  contained  in  21  (tens),  3  (tens)  times,  and  this  3 
we  write  as  the  second  figure  of  the  quotient.  We  com- 
plete the  solution  of  the  example,  as  before  explained. 


DIVISION.  71 

mOBJjJEMS. 

10.  Into  how  many  fields  of  8  acres  each  can  a  farm  of  96 
acres  be  divided  ?  j^^ 

11.  A  glazier  set  966  panes  of  glass  for  a  hotel,  setting  6 
panes  in  each  sash.     How  many  sashes  did  he  use  ? 

12.  How  many  dollars  must  a  man  save  each  year,  to  save 
$968  in  4  years  ?  $^42. 

13.  In  one  season  a  farmer's  wife  made  855  pounds  of  but- 
ter from  the  milk  of  5  cows.  How  many  pounds  of  butter 
was  that  to  each  cow  ?  I7i, 

14.  A  man  bought  a  farm  for  $7,248,  and  paid  for  it  in  4 
equal  yearly  payments.     How  much  did  he  pay  each  year  ? 

15.  In  one  week  there  are  168  hours.    How 

,  .  -,  n  SOLUTION. 

many  hours  m  one  day  ?  of  pkoblem  15. 

16.  How  many  miles  must  I  drive  my  horse        168     7 
each  day,  to  drive  185  miles  in  5  days?    87.  1£_    ^ 

17.  A  livery-man  paid  $288   for   3  horses.  28 
How  much  did  they  cost  him  apiece  ?     $96.  28 

18.  How  many  days  will  it  take  a  girl  to 
braid  1,084  straw  hats,  if  she  braids  4  hats  each  day  ? 

19.  An  iron-founder  sold  1,255  pounds  of  plowpoints, 
weighing  5  pounds  each.    How  many  plowpoints  did  he  sell  ? 

20.  A  coal  dealer  received  $4,627  for  coal,  at  $7  a  ton. 
How  many  tons  did  he  sell  ?  661.  solution 

21.  A  man  built  a  block  of  8  dwellings  at  a      of  problem  21. 
cost  of  $9,888.     How  much  was  the  cost  of       ^^^^ 
each  ?  

22.  In  one  bushel  there  are  4  pecks.     How        J° 
many  bushels  in  6,548  pecks  ?  

23.  At  a  freight  house  are  222  casks  of  kero-         T^ 

sene,  to  be  loaded  on  6  platform  cars.     How         

many  casks  must  be  loaded  upon  each  car  ?  37.  j^| 

24.  What  is  the  quotient  of  44,532  -^  6  ?  — 

25.  How  many  miles  must  a  steam-ship  run  each  day,  to 
make  a  voyage  of  1,771  miles  in  7  days? 

26.  What  is  the  quotient  of  640,955  -J-  7  ?  91,565. 


8 

1236 


FIEST  SOrtmON. 

3248 

8 

32 

406 

4 

0 

48 

48 

72  INTEGERS. 

81.  Ex.  Divide  3,248  by  8. 

Explanation.  —  8  is  contained  in  32 
(hundreds),  4  (hundreds)  times.  We 
write  4  for  the  first  figure  of  the  quotient, 
and  multiplying  by  the  divisor  and  sub- 
tracting, we  have  no  remainder.  The  next 
figure  to  be  divided  is  the  4  (tens)  of 
the  dividend,  and  we  bring  it  down  for  a 
partial  dividend.  Since  8  is  contained  in 
4  (tens),  0  (tens)  times,  we  write  the  0  for  the  second 
figure  of  the  quotient.  8  times  0  (tens)  are  0  (tens), 
and  0  (tens)  subtracted  from  4  (tens)  leaves  4  (tens). 
We  next  bring  down  the  8  (ones),  and  we  have  48  for 
a  new  partial  dividend.  We  complete  the  solution,  as 
before  explained. 

When  we  multiply  the  0  (tens)  of  the  second  solution. 
quotient  by  the  divisor  and  subtract,  the       3248     8 
remainder  thus  obtained  is  the  same  that       32         7^ 
we  had  before.     We  may  therefore  omit  ^g 

this  multiplication    and    subtraction,   as  48 

shown  in  the  second  solution. 

82.  Whenever  the  partial  dividend  is  less  than  the  divi- 
sor, we  place  0  in  the  quotient  and  bring  down  the  next 
figure  of  the  dividend  for  a  new  partial  dividend. 

FMOBIj  EMS. 

27.  How  many  days  will  it  take  a  cooper  to  make  2,718 
barrels,  if  he  makes  9  barrels  each  day  ?  302. 

38.  A  farmer  harvested  536  bushels  of  com  from  8  acres. 
What  was  the  yield  per  acre  ?  67  Imshels. 

29.  In  how  many  days  can  a  teamster  draw  2,187  loads  of 
gravel,  if  he  draws  9  loads  each  day  ? 

30.  Divide  25,041  by  3.  Quotient,  8,347. 

31.  How  much  is  one  sixth  of  $16,236  ?  $2,706. 
33.  If  a  tract  of  1,435  acres  of  land  be  divided  into  7  equal 

faiTns,  how  many  acres  will  there  be  in  each  farm  ? 


DIVISION.  73 

33.  Divide  47,565  by  5.  Qaotimt,  9,513. 

34.  Divide  2,751,075  by  5.  Quotient,  550,215. 

35.  Divide  54,247,296  by  8.  Quotient,  6,780,912. 

36.  What  is  the  quotient  of  78,462,729  -f-  9  ?      8, 718, 081. 

SECOND    METHOD. 

83.  Ex.  What  is  the  quotient  of  4,368  ~  7  ? 

Explanation. — ^We  write  the  dividend  and     soltttiow. 
divisor  as  before,  but  below  the  dividend     43gg  i  7 
we  draw  a  horizontal  line,  under  which  to     ~g24 
write  the  quotient.      7  is  contained  in  43 
(hundreds),  6  (hundreds)  times.     "Wfe  write  the  6  under 
the  horizontal  hne,  directly  below  the  hundreds  of  the 
dividend,  as  the  first  figure  of  the  quotient.     7  times  6 
(hundreds)  are  42  (hundreds).     We  subtract  the  42 
(hundreds)  from  the  43   (hundreds)   of  the  dividend 
mentally,  and  to  the  remainder,  1  (hundred),  we  men- 
talty  unite  the  6  (tens),  making  16  (tens).     7  is  con- 
tained in  16  (tens),  2  (tens)  times.     We  write  the  2 
(tens)  as  the  second  figure  of  the  quotient,  and  multiply 
the  2  (tens)  by  the  divisor  7.     The  result,  14  (tens),  we 
subtract  mentally  from  the  16  (tens),  and  to  the  remain- 
der, 2  (tens),  we  mentally  unite  the  8  (ones),  making 
28.     7  is  contained  in  28,  4  times,  and  this  we  write  as 
the  third  figure  of  the  quotient.     We  again  multiply 
and  subtract  mentally,  and  we  have  no  remainder. 

(See  Manual,  page  217.) 

In  solving  problems  by  this  method,  the  several 
steps  in  the  process  are  the  same  as  by  the  First 
Method;  but  the  multiplications  and  subtractions  are 
performed  mentally,  and  hence  fewer  figures  are  used. 

84.  \Vhen,  in  dividing,  all  the  products  and  partial 
dividends  are  written,  the  process  is  Long  Division. 

85.  When,  in  dividing,  only  the  divisor,  dividend, 
and  quotient  are  written,  the  process  is  Short  Division. 


74  INTEGERS. 

JPMOBLJEMS, 

37.  A  railroad  company  paid  $6,828  for  wood,  at  $6  a  cord. 
How  many  cords  did  they  buy  ?  1^  138. 

38.  How  much  is  1  fourth  of  384  acres  of  land  ? 

39.  If  3  tons  of  hay  will  keep  one  horse  through  the  winter, 
how  many  horses  will  234  tons  keep  ?  78. 

40.  If  a  family  use  8  pounds  of  sugar  in  one  week,  how 
many  weeks  will  416  pounds  last  them  ? 

41.  A  farmer  raised  1,892  bushels  of  turnips  from  4  acres  of 
land.     How  many  bushels  did  the  land  yield  per  acre  ? 

42.  How  many  barrels  of  flour,  at  $8  a  barrel,  can  be  bought 
for  $26,120?  ^  3,265. 

43.  Two  men,  owning  a  ship,  sold  it  to  3  others  for  $13,050. 
How  much  did  each  seller  receive,  and  how  much  did  each 
buyer  pay  ?  Each  luyer  paid  $4,350. 

44.  Divide  386,948  by  4.  Quotient,  96,737, 

45.  Divide  99,627,342  by  6.  Quotient,  16,60^,557, 

46.  What  is  the  quotient  of  8,765,875  -^  5  ? 

47.  What  is  the  quotient  of  200,004,234  ^  6  ? 

48.  Divide  $2,703,848  by  7.  Quotient,  386, 26^. 

49.  The  dividend  is  3,474,963,  and  the  divisor  is  9.  What 
is  the  quotient  ? 

50.  Divide  720,152,088  by  8.  Quotient,  90,019,011. 


CA.SEJ     II. 
The  Divisor  more  than  One  Figure. 

86.  Ex.  Divide  8,058  by  34. 
ExPLAiTATiON. — ^When  the  divisor  con-        solution. 
sists  of  two  or  more  figures,  we  solve  the 
problem  by  long  division.    Since  34  is  con- 
tained in  80,  the  first  two  figures  of  the       125 
dividend,  2  times,  we  place  the  2  for  the      IQ^ 
first  fi^re  of  the  quotient.     2  times  34        238 
are  68,  which  subtracted  from  80  leaves        238 


8058 
68 


34 
237 


DIVISION.  75 

12.  Bringing  down  the  5  of  the  dividend,  we  have  125 
for  a  partial  dividend.  34  is  contained  in  125,  3  times, 
and  we  place  the  3  for  the  next  figure  of  the  quotient. 
3  times  34  are  102,  which  subtracted  from  125,  leaves 
23.  Bringing  down  the  8  of  the  dividend,  we  have  238 
for  a  new  partial  dividend.  34  is  contained  in  238,  7 
times,  and  we  write  the  7  for  the  third  figure  of  the 
quotient.  7  times  34  are  238,  which  subtracted  from 
the  last  partial  dividend,  leaves  no  remainder. 

87  •  Sometimes  we  can  not  tell  exactly  how  many 
times  the  divisor  is  contained  in  a  partial  dividend. 

For  example,  divide  9,796  by  124 

We  do  not  know  how  many  times  124 
is  contained  in  979,  the  first  partial  divi- 
dend, but  we  wiU  suppose  that  it  is  con- 
tained 6  times.  After  multiplying  and 
subtracting,  we  have  a  remainder  of  235, 
which  is  greater  than  the  divisor  124. 
Hence  124  is  contained  in  979  more  than 
6  times.  Let  us  now  suppose  that  it  is 
contained  8  times.  But  8  times  124  is 
992,  which  is  more  than  979.  Hence  124 
is  not  contained  in  979  as  many  as  8  times. 
We  therefore  conclude  that  7  is  the  cor-  ^  ^  ^  ^ 
rect  quotient  figure.  1116 

88.  From  this  example  we  learn  that 

I.  When  any  remainder  is  greater  than  the  divisor,  the 
quotient  figw^e  is  too  small 

n.  When  any  product  is  greater  than  the  partial  divi- 
dend., the  quotient  figure  is  too  great.      oee  Manual,  page  217.) 


FIRST  TEIAL. 

9796 

124 

744 

6 

235 

8KC0ND 

TRIAL. 

9796 

124 

992 

,8 

BOLTJTIC 

N. 

9796 

124 

868 

yf 

76  INTEGERS. 

PM  O  BIj'EMS. 

51.  A  carpenter  received  $156  for  building  12  rods  of  fence. 
How  much  did  he  receive  a  rod  ?  $13. 

52.  How  many  weeks  will  it  take  a  house  painter  to  earn 
$168,  if  he  earns  $14  a  week  ?  12. 

53.  If  a  freight  train  runs  16  miles  an  hour,  how  many 
hours  will  it  be  in  running  368  miles  ?  23. 

54.  In  one  day  there  are  24  hours.     How  many  days  in  816 
hours  ? 

55.  An  oil  well  in  Pennsylvania  produced  5,394  barrels  of 
oil  in  31  days.     How  many  barrels  were  produced  daily  ? 

56.  A  miller  packed  13,475  pounds  of  flour  in  sacks,  put- 
ting 49  pounds  into  each.     How  many  sacks  did  he  fill  ? 

57.  In  Jiow  many  days  can  56  men  do  2,464  days'  work  ? 

58.  A  dealer  received  $1,625  for  sewing-machines,  at  $65 
apiece.     How  many  machines  did  he  sell  ?  25. 

59.  In  a  certain  town  the  bounties  paid  to   93   soldiers 
amounted  to  $78,771.     How  much  was  each  soldier's  bounty  ? 

60.  A  hop  grower  sold  17  bales  of  hops,  which  weighed 
3,536  pounds.     What  was  the  weight  of  each  bale  ? 

61.  A  farmer  harvested  2,520  bushels  of  oats  from  36  acres. 
What  was  the  yield  per  acre  ?  70  hishels. 

62.  A  fish  dealer  packed  1,200  pounds  of  mackerel  in  kits 
holding  25  pounds  each.     How  many  kits  did  he  fill  ?      48. 

63.  A  New  York  daily  newspaper  publisher  uses  96  reams  of 
paper  each  day.     How  many  days  will  10,080  reams  last  him  ? 

64.  In  how  many  days  can  38  men  do  as  much  work  as  one 
man  can  do  in  1,862  days  ?  4^. 

65.  Divide  7,011  by  57.  Quotient,  123. 

66.  Divide  126,378  by  63.  Quotient,  2,006. 

67.  What  is  the  quotient  of  11,952,983  -~  569  ? 

68.  What  is  the  quotient  of  7,000,888  H-  758  ? 

69.  The  dividend  is  4,235,262,  and  the  divisor  1,294.   What 
is  the  quotient  ?  3,273, 

70.  Find  the  quotient  of  20,438,574  -^  4,082. 


DIVISION.  77 

C^SE     III. 
Remainders  after  Dividing  last  Partial  Dividend. 

89t  Ex.  How  many  times  is  7  contained  in  25  ? 
Explanation. — Since  7  is  contained  in    solution. 
21,  3  times,  and  in  28,  4  times  ;  and  siQce    25     7 
21  is  4  less  than  25,  and  28  is  more  than    21     q 
25 ;  7  is  contained  in.  25,  3  times,  with  a    ~i  ^ 

.     t  If  A  4:  Remaiii'ler. 

remamder  oi  4. 


PM  OBI,  JEMS. 

71.  What  will  be  the  remainder,  when  529        solution 

EM   ' 

34 
15 


•        T'lJUO^o  OP  PROBLEM   71. 

IS  dividea  by  84  ? 
72.  How  many  times  is  13  contained  in        34 


217  ?  16  times,  with  a  remainder  of  9:  .  ^.^ 

73.  Into  how  many  farms   of  156   acres        170 
each  can  798  acres  be  divided  ?  ~7Z  „      .  , 

r  *    c  ^  '*%   to  1  J-*  1^  Pwemainder 

Into  5  farms,  with  18  acres  left. 

74.  How  many  dress  patterns  of  13  yards  each  can  be  cut 
from  a  piece  of  mohair  cloth  containing  43  yards  ? 

3,  with  a  remnant  of  J/,  yards. 

75.  What  will  be  the  quotient  and  what  the  remainder, 
when  3,376  is  divided  by  65  ?      Quotient,  51;  remainder,  61. 

76.  The  dividend  is  51,327,  and  the  divisor  807.  Find  the 
quotient  and  the  remainder.      Quotient,  63  ;  remainder,  486. 

77.  A  dairy-man  went  to  an  auction  with  $318,  and  bought 
as  many  cows  at  $48  each  as  he  could  pay  for.  How  many 
cows  did  he  buy,  and  how  many  dollars  had  he  left  ? 

Re  dought  6  cows,  and  had  $30  left. 

78.  How  many  kettles,  each  weighing  348  pounds,  can  be 
made  from  20,000  pounds  of  iron  ? 

57,  and  16^  pounds  of  iron  will  le  left. 

79.  The  dividend  is  246,875,  and  the  divisor  1,159.  What 
is  the  quotient,  and  what  the  remainder  ? 

Quotient,  213 ;  remainder,  8. 

80.  The  dividend  is  705,000,  and  the  divisor  3,275.  Find 
the  quotient  and  the  remainder.  Remainder,  875. 


T8  INTEGERS. 

81.  How  many  sewing- machines  at  $48  each  can  an  agent 
buy  with  $1,185  ?  2Jf,  machines,  and  Trnve  $33  l^t. 

83.  A  silversmith  had  1,096  ounces  of  silver.  After 
making  as  many  silver  pitchers,  each  weighing  45  ounces,  as 
possible,  he  made  a  cake  basket  of  the  silver  he  had  left.  How 
much  did  the  cake  basket  weigh  ?  16  ounces. 

The  Divisor  any  Number  of  Tens,  Hundreds,  and  so  on. 


SOLUTIOJT 

2750 


10 
275 


50 


FIEST  BOLTTTION. 

32764  I  100 
300 


90.  Ex.  1.  Divide  2,750  by  10. 

Explanation. — ^By  this  Solution  it  will  20 

be  seen  that  the  figures  of  the  quotient  75 

are  the  same  as  -  those   of  .  the   dividend  70 
after  removing  the  right-hand  figure. 

Ex.  2.— Divide  32,764  by  100.       _52 

Explanation. — By  the  First  Solution  we 
see  that  if  we  omit  the  two  right-hand 

figures  of  the  dividend,  the  other  figures  ""^"^      I  327 

are  the  same  as  the  quotient ;  and  that  ^^ 

the  two  right-hand  figures  thus  omitted      — 

are  the  same  as  the  remainder.     Hence,  764 

We  may  write  all  of  the  dividend  ex-  '^^ 

cept  the  two  right-hand  figures   for  the  64 

quotient,  and  the  two  right-hand  figures  second  soltjtion. 

for  the  remainder,  as  shown  in  the  second  32764  [  100 

solution.  327     64 

91.  We  may  divide  by  10  by  removing  the  right-hand 
figure  of  the  dividend  ;  by  100  by  removing  the  two  right- 
hand  figures  ;  by  1,000  by  removing  the  three  right-hand 
figures  ;  and  by  any  similar  number,  by  removing  as  many 
figures  from  the  right  of  the  dividend  as  there  are  ciphers 
in  the  divisor. 

92.  The  figures  removed  will  be  the  remainder. 


DIVISION.  79 

83.  At  $10  a  ton  how  much  hay  can  I  buy  for  $450?     43  tons. 

84.  One  day  the  New  York  and  Brooklyn  ferries  received 
87,600  cents  from  passengers.  How  many  dollars  did  they 
receive,  there  being  100  cents  in  one  dollar  ?  $S76. 

85.  An  army  consisting  of  375,000  soldiers  was  divided 
into  regiments  of  1,000  men  each.  How  many  regiments  were 
in  the  army  ?  875. 

86.  Divide  647,500  by  100.  Quotient,  6,475. 

87.  What  is  the  quotient  of  1,627,000-^1,000? 

88.  What  is  the  quotient  of  324,700,000^-10,000  ? 

89.  The  dividend  is  725,000,000  and  the  divisor  100,000. 
What  is  the  quotient  ? 

90.  A  fanner  having  $187,  bought  yearling  cattle  at  $10 
each.  How  many  yearlings  did  he  buy,  and  how  much  money 
had  he  left  ?  He  Ixmglit  18  yearlings,  and  Jiad  $7  left. 

91.  How  many  horses  at  $100  each  can  I  buy  with  $2,765  ? 

/  can  huy  27  horses,  and  have  $65  left. 

92.  What  is  the  quotient  of  76,275-^l,000  ? 

Quotient,  76;  remainder,  275. 

93.  The  dividend  is  32,967,816,  and  the  divisor  10,000. 
What  is  the  quotient,  and  what  the  remainder  ? 

Quotient,  3,296 ;  remainder,  7,816. 


93.  Ex.— Divide  53,485  by  700.  gg^g^ 

Explanation.— By  the  First  Solution  we  ^52^ 

find  the  quotient  to  be  76,  and  the  re-  4485 

mainder  285.  4200 

700  is  100  times  7.  Hence  285 
we  may  first  divide  by  100, 


riEST  soLxmoiT. 

700 


76 


SECOND  BOI-TTTIOJT. 


and    the   quotient  thus    ob-  53485  I  100x7 

tained  by   7.      Dividing   by  

100,  we  have  a   quotient  of  ^    ^^ 

634,  with  a  remainder  of  85.  Quotient   76     2 

Then  dividing  by  7,  we  have  285  Remainder. 


80  INTEGERS. 

a  quotient  of  76,  with  a  remainder  of  2.  This  remainder 
2  prefixed  to  the  first  remainder  85,  makes  285  the 
whole  remainder. 

In  the  Third  Solution  we  first  cut         "^^^^  solution. 
off  by  a  vertical  line  the  ciphers  from       ^5^  1  85  [  7  |  00 
the  divisor,  and  as  many  figures  from       76      285 
the  right  of  the  dividend.     We  then 
divided  the  remaining  figures  of  the  dividend  by  the 
remaining  figures  of  the  divisor  ;  and  for  the  remain- 
der, annexed  85,  the  two  figures  cut  off  fi'om  the  divi- 
dend, to  2,  the  remainder  obtained  by  dividing  by  7. 

94«  ^ules  for  division  of  Integers. 
I.  For  Long  Division. 

1.  Flace  the  divisor  at  the  right  of  the  dividend,  separate 
them  by  a  line,  and  draw  a  line  under  the  divisor  to  sepa- 
rate it  from  the  quotient. 

2.  Find  how  many  times  the  divisor  is  contained  in  the 
first  left-hand  figure  or  figures  of  the  dividend,  and  write 
the  result  under  the  divisor  for  the  first  figure  of  the  quo- 
tient. 

3.  Multiply  the  divisor  by  this  quotient  figure,  and  write 
the  product  under  the  figures  of  the  dividend  already  used. 

4.  Subtract  this  product  from  the  figures  above  it,  and  to 
the  remainder  annex  the  next  figure  of  the  dividend,  thus 
forming  a  partial  dividend. 

5.  Find  how  many  times  the  divisor  is  contained  in  this 
partial  dividend,  and  write  the  result  as  the  second  figure 
of  the  quotient. 

6.  Multiply  the  divisor  by  this  quotient  figure,  subtract 
the  product  from  the  partial  dividend,  and  to  the  remain- 
der annex  the  next  figure  of  the  dividend. 

7.  Proceed  in  the  same  manner  until  all  the  figures  of 
the  dividend  have  been  used. 


DIVISION.  81 

n.  For  Short  Division. 

1.  Write  the  dividend  and  divisor  as  before,  and  draw 
a  line  under  the  dividend  to  separate  it  from  the  quotient. 

2.  Find  how  many  times  the  divisor  is  contained  in  the 
first  left-hand  figure  or  figures  of  the  dividend,  as  in  long 
division,  and  write  the  result  under  the  last  figure  of  the 
dividend  so  used,f  or  the  first  figure  of  the  quotient. 

3.  Multiply,  subtract,  and  form  a  partial  dividend,  as  in 
loug  division,  performing  the  operations  mentally. 

4.  Divide  this  partial  dividend,  and  write  the  result  as 
the  second  figure  of  the  quotient. 

5.  Proceed  in  the  same  manner  until  all  the  figures  of 
the  dividend  have  been  used. 

m.  When  one  or  more  of  the  right-hand  figures  of 
the  divisor  are  ciphers. 

1.  Gut  them  ofiFby  a  line,  and  also  an  equal  number  of 
figures  from  the  right  of  the  dividend. 

2.  Divide  the  remaining  figures  of  the  dividend  by  the 
remaining  figures  of  the  divisor. 

3.  For  the  true  remainder,  annex  to  the  last  remainder 
the  figures  cut  offfrmi  the  dividend. 


SOLUTION  OF  PROBLEM  94. 

24  I  OOP 

12  Quotient. 


PJtOBJDJEMS. 

94.  Divide  299,392  by  24,000. 

95.  A     miller     purchased      9,478  299  |  392 
pounds  of  wheat.     How  many  bushels  24 
did  he  buy,  allowing  60  pounds  to  the  "59 
bushel  ?            157  ImsMs  58  pounds.         48 

96.  In  one  ream  of  paper  there  are      11393  Remainder. 
20  quires.    How  many  full  reams  in 

1,976  quires  ?  98  reams  16  quires. 

97.  What  is  the  quotient  of  387,695  -r-  4,500  ?    What  is  the 
remainder  ?  Remainder,  695. 

98.  How  many  dollars  must  a  man  save  each  year,  to  save 
$2,456  in  8  years?'  $307. 

G 


m 


INTEGERS. 


99.  An  army  of  46,872  men  had  one  tent  for  every  9  sol- 
diers.    How  many  tents  were  in  the  army  ?  5,208. 

100.  What  is  the  quotient  of  3,554,772  -^  124  ? 

101.  What  is  the  quotient  of  746,853  -f-  309  ? 

103.  A  tinsmith  made  1,873  blacking  boxes,  using  1  sheet  of 
tin  for  every  6  boxes.     How  many  sheets  of  tin  did  he  use  ? 

103.  A  pork  buyer  packed  337,600  pounds  of  pork,  putting 
300  pounds  into  a  barrel.     How  many  barrels  did  he  fill  ? 

104.  What  is  the  quotient  of  534,006,487  -j-  9  ?  58,222,9Jt3. 

105.  Divide  919,734,140  by  33,705. 

106.  Divide  503,371,378  by  33,359.  Qmtimt  21,6J,2. 

107.  What  is  the  quotient  of  18,383,959  ~  56,317  ? 

108.  The  dividend  is  340,900,005  and  the  divisor  86,005. 
What  is  the  quotient  ? 

109.  A  drover  paid  $7,194  for  218  head  of  cattle.  What 
was  the  cost  per  head  ? 

110.  How  many  church  bells,  each  weighing  3,245  pounds, 
can  be  made  from  77,880  pounds  of  bell-metal  ? 

111.  An  army  contractor  paid  $39,865  for  2,345  barrels  of 
beef     How  much  did  the  beef  cost  him  per  barrel  ? 

112.  If  272,384  pounds  of  meat  are  distributed  to  9,728 
soldiers  in  a  month,  how  many  pounds  does  each  soldier 
receive  ?  28. 

113.  The  dividend  is  235,073,740  and  the  divisor  43,167. 
What  is  the  remainder  ?  2. 

114.  What  will  be  the  quotient  and  what  the  remainder, 
when  60,190,105  is  divided  by  20,006  ?    Remainder,  12,057. 

115.  If  a  farmer  makes  356  gallons  of  cider,  how  many 
casks  can  he  fill,  putting  40  gallons  into  each  ? 

8  casks,  and  haw  36  gallons  left. 

116.  The  dividend  is  382,775  and  the  divisor  2,500.  What 
is  the  quotient,  and  what  the  remainder  ? 

117.  The  dividend  is  87,693,275  and  the  diviso^  41,700. 
Find  the  quotient  and  the  remainder. 

118.  Divide  8,329,659  by  365,000. 

Qmtient,  22  ;  remainder,  299,659. 


MEASUREMENTS, 


83 


I 
1  Square  Inchrg 


1  incli  long 


SECTION  VI. 

MBA  S  U^BMBJV'TS. 

C^S  E    I, 

Measurement  of  Surface. 

95.  A  surface  1  inch  long  and  1  inch  wide,  having 
square  comers,  is  a  Square  Inch. 
One  that  is  1  foot  long  and  1  foot 
wide  is  a  Square  Foot,  and  one  that 
is  1  yard  long  and  1  yard  wide 
is  a  Square  Yard.  A  Square  Rod 
is  1  rod  long  and  1  rod  wide,  and 
a  Square  Mile  is  1  mile  long  and  1 
mile  wide. 

96.  If  you  have  a  slate  8  inches  long  and  5  inches 
wide,  and  you  draw  lines 
just  one  inch  apart  from  top 
to  bottom,  and  also  from  side 
to  side,  your  slate  will  be  di- 
vided into  squares  each  1 
inch  long  and  1  inch  wide. 
In  each  of  the  5  rows,  count- 
ing from  top  to  bottom,  there 
are  8  square  inches.  Since 
there  are  5  rows  upon  the 
slate,  and  in  each  row  there 
are  8  square  inches,  in  all 
there  are  5  times  8  square 
inches,  or  40  square  inches. 
The  number  of  square  inches 
in  1  row  is  the  same  as  the  number  of  inches  in  the 
length  of  the  slate,  and  the  number  of  rows  is  the  same 
as  the  number  of  inches  in  the  width  of  the  slate. 


84 


INTEGERS 


FMOBjLEMS. 

1.  In  a  village  school  there  is  a  blackboard  16  feet  long 
and  5  feet  wide.  If  lines  be  drawn  upon  it  so  as  to  divide  it 
into  square  feet,  how  many  square  feet  will  there  be  in  one 
row  ?  How  many  rows  will  there  be  ?  How  many  square 
feet  are  there  in  the  blackboard  ?  80  square  feet. 

3.  How  many  square  yards  are  there  in  a  floor  6  yards  long 
and  5  yards  wide  ?  30. 

3.  In  a  pane  of  glass  22  inches  long  and  15  inches  wide 
are  how  many  square  inches  ?  330. 

4.  How  many  panes  of  glass  are  there  in  a  window  that  has 
6  panes  in  a  row,  counting  from  top  to  bottom,  and  4  panes 
in  a  row,  counting  from  side  to  side  ? 

5.  In  a  village  lot  12  rods  long  and  5  rods  wide,  how  many 
square  rods  ?  60. 

6.  How  many  square  miles  in  a  tract  of  land  9  miles  long 
and  7  miles  wide  ? 

7.  How  many  square  feet  in  a  board  16  feet  long  and  1  foot 
wide  ?  16. 

8.  How  many  square  yards  in  the  ceiling  of  a  room  that  is 
10  yards  long  and  7  yards  wide  ? 

9.  How  many  square  miles  in  a 'township  6  miles  square, 
that  is,  6  miles  long  and  6  miles  wide  ?  36. 

10.  A  boy  has  a  checker-board  13  inches  square.  How 
many  square  inches  does  it  contain  ?  IJt^. 

97.  Ex.  A  sheet  of  tin  that  contains  308  square 
inches,  is  22  inches  long.     How  many  inches  wide  is  it  ? 

Explanation. — Since  it  is  22  inches  long, 
in  one  row  there  are  22  square  inches ;  and 
since  in  the  whole  sheet  there  are  308  square 
inches,  there  must  be  as  many  rows  of 
square  inches  as  the  number  of  times  22  is 
contained  in  308,  which  is  14.  There  are  14 
rows ;  and  since  each  row  is  one  inch  wide,  the  whole 
sheet  is  14  inches  wide. 


SOLTTTION. 

308 

22 

22 

14 

88  " 

88 

MEASUREMENTS 


85 


PltOBJLEMS. 

11.  If  a  field  is  44  rods  long,  how  wide  must  it  be  to  con- 
tain 1,584  square  rods  ?  S6  rods. 

12.  A  window  curtain  that  is  3  feet  wide  contains  18  square 
feet.     How  long  is  it  ?  6  feet. 

13.  A  carpenter  used  255  square  feet  of  boards  in  laying  the 
floor  of  a  room  17  feet  long.    How  wide  was  the  room  ? 

14.  Mr.  White's  farm  contains  17,280  square  rods,  and  it  is 
144  rods  long.    How  wide  is  it  ?  120  rods. 


C^SE    II. 
Measurement  of  Capacity. 

98i  A  block,  having  square  comers  and  measuring  1 
inch  each  way,  or  a  box, 
the  hollow  of  which 
is  1  inch  square  and  1 
inch  deep,  contains  a 
cubic  inch.  Any  body 
or  portion  of  space  1 
inch  long,  1  inch  wide, 
and  1  inch  thick  is  a 
Cubic  Inch. 

99.  kCuUc Foot i^l 
foot  long,  1  foot  wide, 
and  1  foot  thick  ;  and 
a  Gvhic  Yard  is  1  yard 
long,  1  yard  wide,  and 
1  yard  thick. 

100.  If  5  blocks,  each  contain- 
ing 1  cubic  inch,  be  placed  side  by 
side,  they  will  form  a  row  5  inches 
long,  1  inch  wide,  and  1  inch 
thick.  If  4  such  rows  be  placed 
side  by  side,   they  will  form  a 


INTEGERS 


layer  of  blocks  5  inches 
long,  4  inches  wide,  and 
1  inch  thick.  As  there 
are  4  rows,  and  in  each 
row  five  blocks,  there 
are  in  the  layer  4  times 
5  blocks,  or  20  blocks. 

In  3  such  layers  there 
are  3  times  20  blocks  or 
60  blocks.  If  we  place 
the  three  layers,  one  ex- 
actly upon  the  other, 
they  will  form  a  pile  5 
inches  long,  4  inches 
wide,  and  3  inches  high. 
There  are  as  many  cubic 
inches  in  one  row  as 
there  are  inches  in  the 
length  of  the  pile,  as 
many  rows  as  there  are 
inches  in  the  width  of 
the  pile,  and  as  many 
layers  as  there  are  inches 
in  the  height  of  the  pile. 


PJt  OBLJEMS. 


15.  Henry  made  a  pile  of  bricks  in  which  there  were  4 
layers,  5  rows  in  each  layer,  and  8  bricks  in  each  row.  How 
many  bricks  were  in  the  pile  ?  160. 

16.  How  many  cubic  inches  in  a  block  9  inches  long,  8 
inches  wide,  and  3  inches  thick  ?  216. 

S  layerSy  8  rows  in  each  loA/er^  and  9  ctiMc  inches  in  each  row. 


MEASUREMENTS.  87' 

17.  A  brick  is  8  inches  long,  4  inches  wide,  and  2  inches 
thick.     How  many  cubic  inches  does  it  contain  ?  Q4' 

18.  A  farmer  has  a  bin  that  is  12  feet  long,  6  feet  wide,  and 
5  feet  deep.     How  many  cubic  feet  does  it  contain  ?       360. 

19.  How  many  cubic  yards  of  earth  will  be  removed  in 
digging  a  cellar  8  yards  long,  6  yards  wide,  and  2  yards  deep  ? 

20.  How  many  cubic  feet  of  stone  will  it  take  to  build  a 
wall  125  feet  long,  6  feet  high,  and  2  feet  thick  ?  1,500. 

21.  In  a  stick  of  timber  24  feet  long,  1  foot  wide,  and  1  foot 
thick,  how  many  cubic  feet  ?  84. 

101.  Ex. — A  boy  who  had  120  blocks,  made  a  pile  of 
them,  putting  8  blocks  in  each  row,  and  3  rows  in  each 
layer.     How  many  blocks  high  was  the  pile  ? 

Explanation.  —  Since  in  1  row  solution. 

there  were  8  blocks,  in  the  3  rows,    ^  ^  ^=^^    j^O     24 

or  1  layer,  there  were  3  times  8  [  5 

blocks,  or  24  blocks.  Since  in  the 
whole  pile  there  were  120  blocks,  and  in  1  layer  24 
blocks,  there  were  in  the  pile  as  many  layers  as  the 
number  of  times  24  is  contained  in  120,  which  is  5.  As 
there  were  5  layers  each  one  block  in  thickness,  the  pile 
was  5  blocks  high. 

JPjROBljUJitS. 

22.  At  another  time,  with  the  120  blocks,  the  boy  made  a 
pile  4  blocks  wide,  and  3  blocks  high.  How  many  blocks 
were  in  the  length  of  the  pile  ?  10. 

23.  He  afterward  made  a  pile  6  blocks  long,  and  4  blocks 
high,  using  all  of  his  blocks.  How  many  blocks  wide  was  the 
pile  ?  5. 

24.  A  block  of  marble  that  contains  6,912  cubic  inches,  is 
48  inches  long  and  18  inches  wide.     How  thick  is  it  ? 

25.  A  mason  has  a  pile  of  stone  that  contains  585  cubic  feet. 
It  is  5  feet  high  and  9  feet  wide.     How  long  is  it  ?     13  feet. 


88  INTEGERS. 

26.  A  farmer  has  in  his  granary  a  bin  which  contains  168 
cubic  feet.  Its  length  is  7  feet,  and  its  width  6  feet.  What  is 
its  depth  ?  j^feet. 

.  27.  In  digging  a  cellar  7  yards  long  and  2  yards  deep,  a 
man  removed  70  cubic  yards  of  earth.  How  wide  was  the 
cellar  ? 

102.  Length,  width,  and  thickness  are  called  Dimen- 
sions. 

103.  A  surface  has  two  dimensions,  length  and 
width. 

104.  A  body  has  three  dimensions,  length,  width, 
and  thickness. 

105.  The  extent  of  any  limited  surface  is  its  Area. 

106.  The  extent  of  any  body  or  portion  of  space  hav- 
ing three  dimensions  is  its  Capacity. 

107.  From  the  explanations  given  in  Cases  I.  and 
n.  of  this  section,  we  deduce  the  following 

General  Principles  of  M^easurement» 

I.  The  area  of  any  surface  having  square  corners  is 
equal  to  the  product  of  its  two  dimensions. 

n.  Either  dimension  of  a  surface  is  equal  to  the  quotient 
obtained  by  dividing  the  area  by  the  other  dimension. 

m.  The  capacity  of  any  body  having  square  corners  is 
equal  to  the  product  of  its  three  dimensions. 

rV.  Any  one  of  the  dimensions  of  such  a  body  is  equal 
lo  the  quotient  obtaiTied  by  dividing  the  capacity  by  the 
product  of  the  other  two  dimensions, 

PItOBJLEMS. 

28.  The  floor  of  a  church  is  96  feet  long  and  40  feet  wide. 
How  many  square  feet  does  it  contain  ?  3,8Ji,0. 

29.  A  farmer,  measuring  his  meadow,  finds  that  it  is  55  rods 
long  and  32  rods  wide.  How  many  square  rods  does  it  con- 
tain? h760. 


PROBLEMS     IN     INTEGERS.  89 

30.  If  48  square  yards  of  oil-cloth  will  cover  the  floor  of  a 
dining-room  8  yards  long,  how  wide  is  the  room  ?     6  yards. 

31.  A  man  used  1,780  square  feet  of  boards  in  building  a 
tight  board  fence  5  feet  high.  What  was  the  length  of  the 
fence?  356  feet. 

82.  I  have  a  box  11  inches  long,  7  inches  wide,  and  3  inches 
deep  on  the  inside.     How  many  cubic  inches  does  it  contain  'i 

33.  A  man  used  896  cubic  feet  of  stone  in  building  a  wall 
4  feet  high  and  2  feet  wide.     How  long  was  the  wall  ? 

34.  If  27  cubic  feet  of  earth  make  one  wagon  load,  how 
many  wagon  loads  will  be  removed  in  digging  a  cellar  33  feet 
long,  24  feet  wide,  and  6  feet  deep  ?  176. 

35.  The  distance  from  one  village  to  another  is  2,304  rods, 
and  the  road  is  4  rods  wide.  How  many  square  rods  of  land 
in  the  road  ?  9,216. 

36.  How  many  square  feet  in  the  roof  of  a  building  32  feet 
long,  the  distance  from  the  ridge  of  the  roof  to  the  eaves  being 
15  feet  on  each  side  ?  960. 

37.  At  10  cents  a  square  foot,  how  many  cents  will  it  cost 
to  build  a  sidewalk  60  feet  long  and  5  feet  ^vide  ?        3,000. 


SECTION    VII. 

1.  Of  a  regiment  of  1,047  men  7  were  regimental  officers, 
and  the  rest  were  divided  into  10  companies.  How  many  men 
were  in  each  company  ?  lOI^. 

2.  How  many  cords  of  wood  at  $6  a  cord  must  be  given  in 
exchange  for  9  barrels  of  flour  at  $8  a  barrel  ?  12. 

3.  A  vessel,  after  sailing  8  miles  an  hour  for  46  hours,  was 
driven  back  272  miles  by  a  storm.  How  far  was  she  then  from 
the  place  of  starting  ?  96  miles. 

4.  In  a  certain  church  26  pews  rent  at  $36  each,  22  at  $24 
each,  and  30  at  $15  each.     For  how  much  do  they  all  rent  ? 

II 


90  INTEGERS. 

5.  A  carpenter  built  me  a  bam  in  7  weeks.  I  paid  Mm  $15 
a  week  for  his  labor,  and  the  lumber  cost  me  $139.  How 
much  did  the  barn  cost  me  ?  $2JtJ^. 

6.  A  grocer  bought  2  cheeses,  one  weighing  76  pounds  and 
the  other  84  pounds,  at  15  cents  a  pound.  How  many  cents 
\y\\\  he  gain  by  selling  both  at  18  cents  a  pound?  JfSO. 

7.  A  farmer  bought  a  lumber  wagon  for  $124,  and  a  markefc 
wagon  for  $140,  and  paid  for  them  in  hay  at  $12  a  ton.  How 
many  tons  of  hay  did  it  take  ?  22. 

8.  A  cattle  dealer  bought  38  head  of  cattle  at  $27  each,  92 
head  at  $31  each,  and  17  head  at  $35  each.  How  much  did 
they  all  cost  him  ?  $J^,J^73. 

9.  A  postmaster  mailed  857  letters  on  Monday,  463  on 
Tuesday,  598  on  Wednesday,  325  on  Thursday,  218  on  Friday, 
and  649  on  Saturday.  How  many  letters  did  he  mail  that 
week?  3,110. 

10.  I  bought  56  acres  of  wood-land  at  $45  an  acre.  After 
selling  the  wood  in  the  tree  for  $1,978,  I  sold  the  laud  at  $20 
an  acre.     Did  I  gain  or  lose,  and  how  much  ?      I  gained  $578. 

11.  A  farmer  had  2  fields  of  barley,  the  first  containing  17 
acres  and  the  second  28  acres.  The  first  yielded  30  bushels 
to  the  acre,  and  the  second  34  bushels.  How  many  bushels  of 
barley  did  he  raise  ?  1,^62. 

12.  A  man  traveled  in  three  successive  days  54  miles,  67 
miles,  and  47  miles.  What  was  the  average  daily ,  distance 
traveled  ? 

108.  Explanation.  —  By  the  Average,  we  solution. 
mean  the  number  of  miles  he  must  have  trav-  5i 

eled  each  day,  to  have  traveled  the  whole  dis-  . ' 

tance  in  the  three  days,  traveling  the  same  dis-  ■ 

tance  each  day.     He  traveled  the  sum  of  54,     168J^ 
G7,  and  47  miles,  or  168  miles  in  3  days.     To  56 

have  traveled  168  miles  in  3  days,  traveling 
the  same  distance  each  day,  in  one  day  be  must  have 
traveled  one  third  of  168  miles  or  56  miles.     56  miles 
was  the  average  daily  distance. 


REVIEW    PROBLEMS.  91 

13.  A  grocer  bought  three  hogsheads  of  molasses  containing 
respectively  135  gallons,  143  gallons,  and  127  gallons.  AVhat 
was  the  average  number  of  gallons  to  a  hogshead  ?         135. 

14.  In  a  village  school  the  number  of  pupils  in  attendance 
on  Monday  was  134,  on  Tuesday  138,  on  Wednesday  143,  on 
Thursday  133,  and  on  Friday  147.  What  was  the  average 
daily  attendance  ?  137  pupils. 

15.  At  a  carpet  manufactory  19,110  yards  of  carpet  were 
woven  in  78  days.  What  was  the  average  number  of  yards 
woven  daily  ?  ^^5. 

16.  A  farmer  fattened  6  hogs  which  weighed  respectively 
312  pounds,  351  pounds,  372  pounds,  395  pounds,  417  pounds, 
and  451  pounds.   What  was  their  average  weight  ?  383 pounds. 

17.  A  merchant's  sales  on  Monday  amounted  to  $348,  on 
Tuesday  to  $317,  on  Wednesday  to  $294,  on  Thursday  to 
$336,  on  Friday  to  $322,  and  on  Saturday  to  $369.  How 
much  were  his  average  daily  sales  ?  $331. 

18.  A  farmer  bought  28  acres  of  land  at  $36  an  acre,  and  35 
acres  at  $27  an  acre.     What  was  the  average  price  per  acre  ? 

19.  A  drover  bought  8  cows  at  $28  each,  and  10  cows  at 
$37  each.     How  much  was  their  average  cost  ?  $33. 

20.  How  many  times  can  114  be  subtracted  from  2,622  ?  23. 

21.  How  many  times  can  I  subtract  500  from  6,575  ? 

13  times,  with  a  final  remainder  of  75. 

22.  A  produce  dealer  sold  28  barrels  of  pork  at  $22  a  bar- 
rel, and  expended  the  money  for  clover  seed  at  $8  a  bushel. 
How  much  clover  seed  did  he  buy  ?  77  hushels. 

23.  52  ladies  and  39  gentlemen  went  on  an  excursion,  and 
their  expenses,  which  were  $3  each,  were  paid  by  the  gentle- 
men.    How  much  did  each  gentleman  have  to  pay  ?        $7. 

24.  The  two  factors  of  a  number  are  87  +  48  and  315  —  142. 
What  is  the  number  ?  ^3, 355. 

25.  What  is  the  difference  between  67  x  83  and  59  +  325  + 
106?  ^^071. 

26.  What  is  the  sum  of  31,253  —  8,494,  127  x  84  and  6,124 
+  3,297?  ^  4^^84S. 


92  INTEGERS. 

27.  A  hardware  merchant  bought  18  sets  of  carriage  springs 
weighing  60  pounds  each,  29  sets  weighing  37  pounds  each, 
and  64  sets  weighing  45  pounds  each.  How  many  sets  did  he 
buy,  and  how  much  did  they  all  weigh  ? 

They  all  weighed  5, 033  pounds. 

28.  A  grocer  commenced  business  with  $2,500.  The  first 
year  he  gained  $687,  and  the  second  year  he  lost  $1,428.  The 
next  four  years  his  average  yearly  gain  was  $863.  How  much 
was  he  worth  at  the  end  of  the  six  years  ?  $5,211. 

29.  If  I  buy  137  acres  of  land  at  $64  an  acre,  and  pay  out 
$876  for  fences  and  buildings  upon  it,  how.  much  does  it  cost 
me?  $9,6U' 

30.  A  man  buys  a  farm  for  $7,865,  agreeing  to  make  one 
payment  each  year,  each  payment  except  the  last  to  be  $500. 
How  many  payments  will  he  have  to  make,  and  what  will  be 
the  last  payment  ?  16  payments^  the  last  one  $365. 

31.  A  coal  dealer  bought  350  tons  of  coal,  receiving  2,240 
pounds  for  a  ton.  He  sold  it  at  2,000  pounds  for  a  ton. 
How  many  tons  did  he  sell  ?  392. 

32.  A  grocer  bought,  at  different  times,  198  pounds  of  but- 
ter, 57  pounds,  324  pounds,  96  pounds,  696  pounds,  and  197 
pounds.  He  packed  it  in  tubs,  putting  56  pounds  into  each. 
How  many  tubs  did  he  fill  ?  28. 

33.  A  man  has  a  lot  60  feet  front  and  126  feet  deep,  but  as 
it  is  low  he  wishes  to  fill  it  in  2  feet.  How  many  wagon  loads 
of  earth  will  be  required,  27  cubic  feet  being  one  load  ?      560. 

34.  A  man  paid  3  cents  a  day  for  the  schooling  of  each  of 
his  3  children.  The  children  were  in  school  5  days  each 
week  for  42  weeks  in  the  year.  How  much  did  their  year's 
schooling  cost  him  ?  1,890  cents. 

35.  The  same  man  expended  for  cigars  6  cents  a  day  for 
each  of  the  365  days  of  the  year.  Which  cost  him  the  more,  his 
cigars  or  the  schooling  of  his  children  ?     How  much  more  ? 

His  ciga/rs  cost  300  cents  nmre. 

36.  A  young  man  worked  a  year  at  $25  a  month.  He  paid 
$8  a  month  for  board,  and  expended  $100  more.  How  much 
money  did  he  save  ?  •  $10J!f. 


REVIEW    PROBLEMS.  93 

37.  What  is  the  average  weight  of  8  bales  of  cotton  which 
weigh  respectively  385,  367,  418,  374,  396,  405,  373,  and 
403  pounds  ?  390  pounds. 

38.  A  manufacturer  made  3,468  yards  of  cloth,  and  sold 
1,383  yards  for  $4,146,  and  the  remainder  for  $4,344.  How 
much  did  he  receive  per  yard  for  each  of  the  two  lots  ? 

$3  a  yard  for  tJieJirst,  and  $4.  a  yard  for  the  second. 

39.  A  merchant  bought  17  pieces  of  alpaca,  each  piece  con- 
taining 43  yards.  After  selling  140  yards,  how  many  dress 
patterns,  of  14  yards  each,  had  he  left  ?  ^1. 

40.  If  I  buy  a  mill  for  $13,675,  and  pay  down  $1,675,  how 
many  payments  of  $1,375  each  shall  I  have  to  make  to  pay 
the  balance.  8. 

41.  Harry  is  6  years  old,  and  his  grandmother  is  10  times  as 
old  as  he.  Should  they  both  live  until  he  is  9  years  old,  how 
many  times  as  old  as  Harry  will  his  grandmother  then  be  ?  7. 

43.  At  a  cotton  mill  1,071,399  yards  of  cloth  were  made  in 
313  days.     How  many  yards  were  made  daily  ?  3^1^23. 

43.  In  the  manufacture  of  this  cloth  357,133  pounds  of 
cotton  were  used.     How  many  pounds  were  used  daily  ? 

44.  How  many  yards  of  cloth  were  made  from  each  pound 
of  cotton  ?  3. 

45.  A  farmer  had  three  flocks  of  sheep,  the  first  containing 
184,  the  second  318,  and  the  third  65  sheep.  He  sold  114 
sheep  from  the  first  fiock,  189  from  the  second,  and  48  from 
the  third.  How  many  sheep  did  he  sell,  and  how  many  had 
he  left  ?  He  had  116  sheep  left. 

46.  A  railroad  embankment  is  1,300  yards  long;  its  aver- 
age width  is  38  yards,  and  its  average  height  10  yards.  How 
many  cubic  yards  of  earth  does  it  contain  ?  36 Jf^  000. 

47.  If  3  bushels  of  onions  can  be  raised  from  1  square 
rod  of  ground,  how  many  bushels  can  be  raised  in  a  garden  13 
rods  long  and  9  rods  wide  ?  216. 

48.  What  is  the  quotient  of  14  times  396  divided  by  33 
times  43  ?  ^. 

49.  From  3,738  +  5,393  +  137,396  subtract  3,379,  and  di- 
vide the  remainder  by  193.  Quotient  736, 


CHAPTER  II. 
DECIMALS. 

SECTION  I. 
■'''    y^  X     y    ■■''    .'-'"'  -« 


pnc 


T\yo 


lif^iiiiii:»iiiiii 


lljFI!!l!l^l^i!^i^ii!l|iiiii^l!^ 


109.  If  we  divide  a  block  into  10  equal  parts,  1  of 
the  parts  is  1  tenth  of  the  whole  block,  2  of  the  parts 
are  2  tenths,  3  of  the  parts  are  3  tenths,  and  4  of  the 
parts  are  4  tenths.  In  the  whole  block  there  are  10 
tenths. 

When  any  thing  or  number  is  divided  into  10  equal 
parts,  1  of  the  parts  is  /  tenth  of  the  thing  or  number, 
8  of  the  parts  are  3  tenths,.  5  of  the  parts  5  tenths,  6  of 
the  parts  6  tenths,  7  of  the  parts  7  tenths,  and  so  on. 

1  tenth  is  written  .1 

2  tenths  are  written  .2  6  tenths  are  written  .6 

3  "         "         "         .S 

4  "        ''        "        4 

5  "        "         "        .5 

110.  The  number  111  consists  of  1  hundred  1  ten 
and  1  one.  Since  in  1  hundred  there  are  10  tens,  the 
1  ten  is  1  tenth  of  1  hundred  ;  and  since  in  1  ten  there 
are  10  ones,  the  1  one  is  1  tenth  of  1  ten.     Hence, 

The  value  of  any  figure  in  a  number  is  1  tenth  of  the 
value  of  a  like  figure  standing  in  the  next  place  at  the  left. 


7 

u 

u 

(( 

.7 

8 

u 

a 

a 

.8 

9 

u 

u 

a 

.9 

NOTATION    AND    NUMERATION. 


95 


111.  The  value  of  a  figure  written  at  the  right  of 
ones  is  1  tenth  as  great  as  the  value  of  a  like  figure  in 
the  ones'  place,  and  therefore  it  is  tenths. 

11  and  1  tenth   is  written  11.1 

5  and  3  tenths  "       "  5.3 

16  and  5  tenths  "       "         16.5 

510  and  6  tenths  "       "       510.6 

112.  The  period  or  point  which  is  placed  before 
tenths  is  the  Decimal  Point.  The  figure  at  the  left  of 
the  decimal  point  is  always  ones,  and  the  figure  at  the 
right  of  it  is  always  tenths. 

When  the  decimal  point  stands  between  figures,  it  is 
read  and.  Thus,  5.2  is  read  5  and  2  tenths ;  290.4  is 
read  290  and  4  tenths. 

JEXEM  CIS  JE8. 

1.  Bead  .3,  .9,  .5;  5.2,  18.1,  40.6. 

2.  Write  3  and  5  tenths. 
Write  6  and  4  tenths. 

Write  8  tenths.  |      5.  Write  124  and  7  tenths. 

Write  50  and  5  tenths. 
Write  32700  and  1  tenth. 


3. 

4. 
6. 
7. 
8.  Read  7.1,  65.8,  393.3,  200.9,  2363.4. 


113.  If  a  board  10  inches 
square  be  divided  into  strips 
1  inch  wide,  each  strip  will 
be  1  tenth  of  the  whole 
board.  If  each  of  these 
strips  be  divided  into  10 
equal  parts,  in  the  10  strips, 
or  the  whole  board,  there 
will  be  10  times  10  or  100 
equal  parts.  1  of  these  parts 
will  therefore  be  1  hundredth 
of  the  board. 


j 

!  !  M  ]  r 

i 

i    J     i    J     L   i 

i 

i    1    i    i    >•■  j 

i 

-'    i    I    !    11 

...J-.. 

— 

.-J..J.J...JJ...J... 

i   i    M   i    1 

—  j— 

.-- 

i     i     i     1     i     r    ■ 

_     1       '       1       I       1       ; 



io 

n 

e|T|e!njt  ihl 

96  DECIMALS. 

"When  1  tenth  of  any  thing  or  number  is  divided  into 
10  equal  parts,  each  part  will  be  1  hundreth  of  the  whole 
thing  or  number. 

114.  Since  the  value  of  a  figure  in  any  place  is  1 
tenth  of  the  value  of  a  Hke  figure  in  the  next  place  at 
ihe  left,  a  figure  written  at  the  right  of  tenths  must  be 
Jiundreths.     Thus, 

.11  is  one  tenth  and  1  hundredth. 
.83  is  8  tenths  and  3  hundredths. 
3.27  is  3  ones,  3  tenths,  and  7  hundredths. 
.05  is  0  tenths  and  5  hundredths. 

115.  .35  is  3  tenths  and  5  hundredths  ;  but  3  tenths 
=  30  hundredths,  and  30  hundredths  +  5  hundredths 
=  35  hundredths. 

Tenths  and  hundredths  are  read  together  as  hundredths. 

.52  is  5  tenths  and  2  hundredths,  and  is  read  52  hun- 
dredths. 
.96  is  96  hundredths. 
.03  is  3  hundredths. 
5.24  is  5  and  24  hundredths. 
298.05  is  298  and  5  hundredths. 

EXEJtCISES, 

9.  Read  11.18,  10.24,  81.6. 

10.  Read  40.93,  128.52,  50.07. 

11.  Read  7.08,  217.01,  3000.02. 

12.  Write  7  tenths  and  3  hundredths,  or  73  hundredths. 

13.  Write  5  tenths  and  1  hundredth,  or  51  hundredths. 

14.  Write  27  hundredths ;  3  hundredths. 

15.  Write  4  and  15  hundredths ;  4  and  5  hundredths. 

16.  Write  800  and  21  hundredths. 

17.  Write  18000  and  1  hundredth. 

116.  A  figure  at  the  right  of  hundredths  is  thou- 
sandths ;  and  tenths,  hundredths,  and  thousandths  are 
read  together  as  thousandths. 


NOTATION     AND     NUMERATION.  97 

.456  is  4  tenths,  5  hundredths,  and  6  thousandths,  and  is 

read  456  thousandths. 
.209  is  2  tenths,  0  hundredths,  and  9  thousandths,  and  is 

read  209  thousandths. 
.063  is  read  63  thousandths. 
.004  is  read  4  thousandths. 
3.528  is  read  3  and  528  thousandths. 
80.082  is  read  80  and  82  thousandths. 

117.  Numbers  expressed  by  ones,  tens,  hundreds, 
etc.,  are  Integers  or  Whole  Numbers. 

118.  Numbers  expressed  by  tenths,  hundredths, 
thousandths,  etc.,  are  Decimals. 

119.  A  number  consisting  of  an  integer  and  a  deci- 
mal is  a  Mixed  Number. 

120.  In  writing  decimals  and  mixed  numbers  the 
decimal  point  must  always  be  used. 

exeh  CIS es. 

18.  Read  .275,  7.463,  32.416. 

19.  Read  86285.419,  .507,  700.256. 

20.  Read  11.092,  .048,  .002. 

21.  Read  214.005,  .001,  217.908. 

22.  "Write  5  and  376  thousandths. 

23.  Write  3250  and  615  thousandths. 

24.  "Write  43  thousandths ;  81  thousandths. 

25.  "Write  87  and  87  thousandths. 

26.  Write  401  thousandths ;  7  thousandths. 

27.  Write  9000  and  9  thousandths. 

28.  Write  18  and  305  thousandths. 

29.  Write  101000  and  101  thousandths. 

121.  A  figure  at  the  right  of  thousandths  is  ten-thou- 
mndths;  and  a  decimal  containing  tenths,  hundredths, 
thousandths,  and  ten-thousandths  is  read  as  ten-thou- 
sandths. 

.2574  is  2  tenths,  5  hundredths,  7  thousandths,  and  4  ten- 
thousandths,  and  is  read  2574  ten-thousandths. 


98  DECIMALS. 

.0452  is  452  ten- thousandths. 
6.0048  is  6  and  48  ten-thousandths, 
59.0006  is  59  and  6  ten-thousandths. 
6000.3001  is  6000  and  3001  ten- thousandths. 
204.0809  is  204  and  809  ten-thousandths. 

122.  A  figure  at  the  right  of  ten-thousandths  is 
hundred-thousandths.  Thus  .32516  is  3  tenths,  2  hun- 
dredths, 5  thousandths,  1  ten-thousandth,  and  6  hun- 
dred-thousandths, and  is  read  32516  hundred-thou- 
sandths. 

A  figure  at  the  right  of  hundred-thousandths  is 
millionths.     Thus  .259361  is  259361  mmionths. 

A  figure  at  the  right  of  millionths  is  ten-millionthSy 
and  a  figure  at  the  right  of  ten-miUionths  is  hundred- 
millionths. 

.63015  is  63015  hundred-thousandths. 

.40003  is  40003  hundred -thousandths. 

.029304  is  29304  millionths. 

.000007  is  7  millionths. 

.2367592  is  2367592  ten-millionths. 

.0000024  is  24  ten-millionths. 

.59642108  is  59642108  hundred-millionths. 

.00000009  is  9  hundred-millionths. 


>*  TABLE  OF  VALUES  OF  DECIMAL  NUMBERS. 

One  decimal  figure  expresses  tenths. 
Two  decimal  figures  express  liundredtJis. 


Three 

u 

"        thousandths. 

Four 

u 

"        ten-thousandths. 

Five 

u 

"        hundred-thousandths. 

Six 

(( 

"        millionths. 

Seven 

u 

"        ten-millionths. 

Eight 

u 

u 

"        hundred-millionths. 

(See  Manual,  page  218.) 

NOTATION    AND    NUMERATION. 


99 


124.  Figures  standing  in  places  at  equal  distances  to 
the  right  and  left  of  ones  have  naines  that  correspond 
to  each  other,  as  shown  in  the  following 


DIAGRAM    OF    DECIMAL    NOTATION. 

765432    1.   234567 


8     0 


125(  The  place  which  any  decimal  figure  occupies  in  a 
number  determines  the  value  expressed  by  it  in  that 
number. 

126.  The  relative  values  of  the  different  places  in 
decimals  are  shown  by  the  following 


DECIMAL     NOTATION     AND     NUMERATION     TABLE 


1  one 

1  tenth 

1  hundredth 

1  thousandth 

1  ten-thousandth 

1  hundred-thousandth 

1  millionth 

1  ten-millionth 

10  tenths 

10  himdredths 

10  thousandths 

10  ten-thousandths 

10  hundred-thousandths 

10  millionths 

10  ten-milHonths 

10  hundred-millionths 


is  10  tenths. 

"  10  hundredths. 

"  10  thousandths. 

"  10  ten-thousandths. 

"10  hundred-thousandths. 

"  10  millionths. 

"  10  ten-millionths. 

"  10  hundred-millionths. 

are  1  one. 
"     1  tenth. 
".    1  hundredth. 
"     1  thousandth. 

1  ten-thousandth. 

1  hundred-thousandth. 

1  millionth. 

1  ten-millionth. 

(See  Manual,  page  218.) 


100  DECIMALS. 

127.  Since  the  value  of  every  figure  in  a  decimal  is 
determined  by  the  place  it  occupies,  and  since  ciphers 
on  the  right  of  a  decimal  do  not  change  the  places  of 
the  other  figures  in  the  number,  it  follows  that 

I.  Ciphers  may  he  annexed  to  any  decimal,  or  decimal 
ciphers  to  any  integer,  without  changing  its  value  ;  and 

II.  Ciphers  may  be  omitted  from  the  right  of  any  deci- 
mal, or  decimal  ciphers  from  the  right  of  any  integer,  with- 
out changing  its  value.      (See  Manual,  page  218.) 

JEXEMCISES. 

„^    ^      ,  .  (See  Manual,  page  218.) 

30.  Read  .523,  4.376,  .009,  .027,  .209. 

31.  Read  9.018,  100.003,  435.125. 

32.  Read  .2987,  4.0232,  18.0901,  .1805,  14.0029. 

33.  Read  365.0007,  60.1273,  400.5017. 

34.  Read  .7025,  .7005,  .0005,  30.6008. 

35.  Read  .24731,  .09671,  .10006,  .00008. 

36.  Read  .00055,  .70438,  8.52804. 

37.  Read  .90052,  250.46031,  349.30116,  1000.20084. 

38.  Read  .256153,  709.400365,  .4366576. 

39.  Read  .1413948,  32876.2850041,  25.530016. 

40.  Read  217.1800624,  59.00654387,  3054.26405746. 

41.  Read  32957251.5283563,  172000650.50040036. 

42.  Write  sixteen  ten-thousandths. 

43.  Write  five  hundred  seventeen  and  three  thousand  six 
hundred  forty-seven  ten-thousandths. 

44.  Write  thirty-six  thousand  two  hundred  seventy-three 
hundred-thousandths. 

45.  Write  fourteen  hundred-thousandths. 

46.  Write  five  thousand  eighteen  hundred-thousandths. 

47.  Write  two  hundred  seventeen  thousand  four  hundred 
fifty-six  millionths. 

48.  Write  six  hundred  thousand  two  hundred  eighty-four 
millionths. 

49.  Write  one  hundred  ninety-three  millionths. 


ADDITION.  101 

50.  "Write  sixteen  million  three  hundred  fifty-eight  thousand 
seven  hundred  twenty-four  ten-millionths. 

51.  Write  forty-six  million  two  hundred  seventy-four  thou- 
sand five  hundred  eight  hundred-millionths. 

52.  Write  78,319  ten-millionths. 

53.  Write  6  and  49  hundi'ed-millionths. 

54.  Write  106,204  hundred-millionths. 

55.  Write  7,017  and  4  millionths. 

56.  Write  4  and  68,001  ten-millionths. 

57.  Write  44  and  44  hundred-thousandths. 

58.  Write  975  million  206  thousand  410  and  3  tenths. 

59.  Write  50  million  and  50,512  hundred-millionths. 

60.  Write  101  million  101  thousand  101  and  1,001,001  hun- 
dred-millionths. 


SECTION    II. 
A.  S)  ^  I  T I  O  JV. 

128.  Ex.  What  is  the  sum  of  45.75,  29.36,  and  442? 

Explanation. — ^We  write  tens  under  tens,  soLirrioN, 
ones  under  ones,  tenths  under  tenths,  and  45.75 
hundredths  under  hundredths.  The  decimal  29.36 
points  then  stand  in  a  column.  We  commence  ^-^^ 
at  the  right,  and  add  as  in  integers.  The  sum  79.53 
of  the  column  of  hundredths  is  hundredths, 
and  the  sum  of  the  column  of  tenths  is  tenths.  We 
must  therefore  place  the  decimal  point  at  the  left  of  the 

5  in  the  sum.      (See  Manual,  page  219.) 

129.   The  decimal  point  in  the  sum  must  always  he 
placed  directly  below  the  decimal  points  in  the  parts. 


102  DECIMALS. 

JPMOB  LEMS. 

1.  A  farmer  had  98.7  acres  of  land,  and  bought  15.5  acres 
more.     How  many  acres  had  he  then  ?  IIJ^.2. 

2.  A  lady  bought  two  carpets,  the  first  contaming  27.5  yards, 
and  the  second  23.5  yards.  How  many  yards  of  carpeting  did 
she  buy  ?  51  yards. 

3.  A  farmer  who  had  three  meadows  cut  from  the  first  18.68 
tons  of  hay,  from  the  second  15.27  tons,  and  from  the  third  13.54 
tons.     How  much  hay  did  he  cut  from  the  three  meadows  ? 

4.  A  silversmith  used  3.65  ounces  of  silver  in  making  a  set 
of  tea-spoons,  8.72  ounces  for  a  set  of  table-spoons,  and  11.63 
ounces  for  a  set  of  forks.  How  much  silver  did  he  use  for 
all  ?  ,  ^Jf.  ounces. 

5.  A  gentleman  has  three  village  lots,  one  of  which  contains 
.8  of  an  acre,  another  .5,  and  the  third  3.4  acres.  How  many 
acres  has  he  in  the  three  lots  ? 

6.  A  housekeeper  burned  .728  of  a  ton  of  coal  in  December, 
.835  of  a  ton  in  January,  .694  of  a  ton  in  February,  and  .532 
of  a  ton  in  March.  How  many  tons  did  she  burn  in  the  four 
months  ?  2.789. 

7.  At  a  Pennsylvania  iron  mine  216.845  tons  of  iron  were 
mined  on  Monday,  204.376  tons  on  Tuesday,  198.275  tons  on 
Wednesday,  220.615  tons  on  Thursday,  187.945  tons  on  Friday, 
and  206.004  tons  on  Saturday.  How  many  tons  were  mined 
in  the  week  ?  1234.06. 

8.  A  wood  dealer  has  on  hand  57.75  cords  of  hickory 
wood,  139.75  cords  of  maple  wood,  67.65  cords  of  beech 
wood,  and  78.26  cords  of  hemlock  wood.  How  many  cords 
of  wood  has  he  ?  34S.4I. 

9.  A  farmer  has  four  meadows.  The  first  contains  10.15 
acres,  the  second  9.76  acres,  the  third  12.25  acres,  and  the 
fourth  7.82  acres.  How  much  land  has  he  in  the  four 
meadows  ?  S9.98  acres. 

10.  The  cargo  of  a  coal  barge  consisted  of  45.75  tons  of 
chestnut  coal,  69.54  tons  of  stove  coal,  36.94  tons  of  egg  coal, 
and  51.25  tons  of  lump  coal.  What  was  the  whole  amount  of 
coal  in  the  cargo  ?  203 4S  tons. 


ADDITION.  103 

11.  A  man  traveled  125.75  miles  by  stage,  313.75  miles  by 
railroad,  and  89.45  miles  by  steamboat.  How  many  miles  did 
lie  travel  in  all  ?  428.95. 

13.  A  tavern  keeper  bouglit  four  loads  of  hay,  the  first  con- 
taining 1.156  tons,  the  second  1.328  tons,  the  third  .987  tons, 
and  the  fourth  1.048  tons.     How  much  hay  did  he  buy  ? 

13.  What  is  the  sum  of  1.325  +  .865  +  .655  ?  S.745. 

130.  Ex.  AMiat  is  the  sum  of  7.4675,  836.5,  85.275, 
and  973  ? 

Explanation. — After  writing  the  num-  solution.  . 

bers  with  the  decimal  points  in  a  column,  7.4675 

thus  bringing  ones   under   ones,   tenths  q^  97^0 

under  tenths,  etc.,  we   annex  ciphers  to  973  0000 

the   second,  third   and  fourth  numbers       ■ ' 

until  each  contains  as  many  decimal  fig-  1902.^4^5 
ures  as  the  first  number.  (See  127.)  We  then  add  as 
in  integers,  and  place  the  decimal  point  in  the  sum 
directly  under  the  decimal  points  in  the  parts. 

JPU  OBLJEMS. 

14.  A  tailor  bought  two  pieces  of  cassimere,  one  containing 
08.5  yards,  and  the  other  41.35  yards.  How  many  yards  of 
cloth  in  both  pieces  ?  79.75. 

15.  One  hour  a  railroad  train  ran  33.6  miles,  the  next  hour 
38.83  miles,  and  the  third  hour  33.37  miles.  How  far  did  it 
mn  in  the  three  hours  ?  7^.69  miles. 

16.  In  four  successive  weeks  an  ice  dealer  sold  18.363  tons, 
15  967  tons,  17.4  tons,  and  16.48  tons  of  ice.  How  much  ice 
•lid  he  sell  in  the  four  weeks  ?  68.11  tons. 

17.  How  many  tons  of  straw  are  there  in  five  stacks,  the 
first  of  which  contains  3.7  tons,  the  second  4.1375  tons,  the 
third  3.875  tons,  the  fourth  5.3  tons,  and  the  fifth  7.65  tons  ? 

23.5525. 

18.  A  fruit  dealer  bought  five  lots  of  cranberries.  The  first 
lot  contained  5.75  bushels,  the  second  lot  8.5  bushels,  the  third 
lot  6.635  bushels,  the  fourth  lot  9  bushels,  and  the  fifth  lot 
4.35  bushels.     How  many  bushels  were  there  in  the  five  lots  ? 


104  DECIMALS. 

131«   llute  for  Addition  of  decimals. 

I.  Write  the  numbers  so  that  the  decimal  points  shall 
stand  in  a  column, 

II.  Add  in  the  same  manner  as  in  integers^  and  place 
the  decimal  point  in  the  sum  directly  under  the  decimal 
points  in  the  parts. 

PJROBLEMS. 

.  19.  What  is  the    sum    of   5.0084  +  641.385  +  9.00843  + 
21.000001  +  5.064  ?  681465831. 

20.  A  merchant  sold  1.75  bushels  of  clover  seed  to  one 
farmer,  3  bushels  to  another,  2.5  bushels  to  a  third,  4.225 
bushels  to  a  fourth,  and  3.25  bushels  to  a  fifth.  How  many 
bushels  did  he  sell  to  the  five  farmers  ?  U.725. 

21.  What  is  the  sum  of  five  and  five  tenths,  five  and  five 
hundredths,  eight  and  seventy-five  thousandths,  twenty-one 
and  three  thousand  six  ten-thousandths,  and  five  thousand 
nine  and  six  hundred  forty  thousand  seventeen  millionths  ? 

50J^9.565617. 

22.  A  gardener  sold  4.5  bushels  of  beans  to  one  grocer,  7 
bushels  to  another,  3.25  bushels  to  a  third,  1.625  bushels  to  a 
fourth,  and  2.125  bushels  to  a  fifth.  How  many  bushels  did 
he  sell  to  all  ?  18.5. 

23.  What  is  the  sum  of  three  hundred  and  three  hundredths, 
one  thousand  seven  and  two  hundred  thousand  six  millionths, 
one  hundred  seventeen  thousand  seven  hundred  nine  and  six 
hundred  four  ten-thousandths,  and  eight  and  fifty-two  mil- 
lionths ?  1190^.290458. 


(24) 

(25) 

(26) 

(27) 

47.25 

.967 

.125 

8000.1 

5.00695 

.00054 

1.25 

96.2006 

193.9 

953.5 

12.5 

504.40307 

5.876 

7.375 

.0125 

2046.25 

94.376964 

1000.0001 

.00125 

9.0004 

9.00005 

6.75 

7.3 

28.4 

290.050063 

8.80808 

.0827 

167.283 

483 

.000006 

5.5008 

SUBTKACTION.  105 

SECTION   III. 

S  ZrS  T^A.  C  TIOJV, 

132.  Ex.  From  38.25  subtract  16.78. 

Explanation. — ^We  write  the  numbers  so  solution. 
that  the  decimal  point  of  the  subtrahend  shall  ?5*Ho 
stand  directly  under  that  of  the  minuend,  and  — 'J— 
subtract  as  in  integers.  Since  hundredths  21.47 
subtracted  from  hundredths  leave  hundredths, 
and  tenths  subtracted  from  tenths  leave  tenths,  there 
are  hundreths  and  tenths  in  the  remainder  ;  we  there- 
fore place  the  decimal  point  before  the  4. 

133*  The  decimal  point  in  the  remainder  must  always 
he  directly  under  the  decimM  point  in  the  minuend  and 
subtrahend. 

PMOBIiEMS. 

1.  A  merchant  sold  16.7  yards  of  calico  from  a  piece  that 
contained  33.4  yards.  How  many  yards  remained  in  the 
piece  ?  16.7. 

2.  A  farmer  raised  33.25  bushels  of  clover  seed,  and  sold 
24.75  bushels.     How  many  bushels  had  he  left  ?  7.5. 

3.  A  druggist  bought  a  cask  of  wine  containing  36.5  gal- 
lons. After  selling  19.5  gallons,  how  much  remained  in  the 
cask  ?  17  gallons. 

4.  At  night  the  ice  on  the  river  was  7.37  inches  thick,  and 
in  the  morning  it  was  9.03  inches  thick.  How  much  ice  had 
formed  during  the  night  ?  1.66  inches  thick. 

5.  A  wood  dealer  bought  398.65  cords  of  wood,  and  sold 
276.25  cords.     How  many  cords  remained  unsold  ?      122.^. 

6.  Of  a  railroad,  which  is  to  be  156.325  miles  long  when 
finished,  83.875  miles  are  built.  How  many  miles  remain 
unfinished  ?  72.J^5. 

7.  Mr.  West's  farm  is  116.36  rods  in  length  and  29.28  rods 
less  in  width.     What  is  its  width  ?  87.08  rods. 

I 


106  DECIMALS. 

134.  Ex.  From  130.5  subtract  93.1875. 
Explanation. — Since  cipliers  may  be  an-         solution. 
nexed  to    any  decimal    number  without        130.5000 
changing  its  value,  (see  127),  we   annex         93.1875 
ciphers  to  the  minuend  until  it  contains  as  37.3125 

many  decimal  figures  as  the  subtrahend, 
and  then  subtract,  and  place  the  decimal  point  as  be- 
fore explained. 

PM  OBIjJEMS. 

8.  From  a  cask  containing  31.5  gallons  of  vinegar  20.75 
gallons  were  drawn.  How  many  gallons  remained  in  the 
cask  ?  10.75. 

9.  A  man  put  into  his  wood-house  24.5  cords  of  wood  for 
his  year's  supply,  and  at  the  end  of  the  year  he  had  2.875 
cords  left.     How  many  cords  had  he  used  ?  21.625. 

10.  Two  men  built  134  rods  of  stone  fence,  one  of  them 
building  65.87  rods.     How  many  rods  did  the  other  build  ? 

11.  From  a  hogshead  of  molasses  containing  135.5  gallons, 
1.175  gallons  leaked  out.  How  many  gallons  were  then  in 
the  hogshead  ?  ^  134.325. 

12.  A  farmer  having  217.625  acres  of  land,  sold  87.0375  acres. 
How  much  land  had  he  left  ?  130.5875  acres. 

135.   ^tcle  for  Subtraction  of  decimals, 

I.  Write  the  numbers  with  the  decimal  point  of  the  sub- 
trahend directly  under  that  of  the  minuend. 

n.  Subtract  in  the  same  manner  as  in  integers,  and 
place  the  decimal  point  in  the  remainder  directly  under  the 
decimal  point  in  the  subtrahend. 

PMOJil^  EMS. 

13.  From  a  box  that  contains  10  pounds  of  indigo  a  grocer 
sold  7  0625  pounds.  How  many  pounds  remained  in  the 
box?  ^'^^'^• 

14.  From  ninety-five  and  forty-four  thousandths  take  ten 
and  eight  thousand  five  ten-thousandths. 


SUBTRACTION.  107 

15.  A  liberty  pole  83.5  feet  long  was  set  in  the  ground  8.75 
feet.  How  many  feet  from  the  ground  to  the  top  of  the 
pole?  7Jt.75 

16.  A  cubic  inch  of  silver  weighs  6.061  ounces,  and  a  cubic 
inch  of  marble  1.641  ounces.  How  much  heavier  is  the  silver 
than  the  marble  ?  44^  ounces. 

17.  923.85  miles  —  385.275  miles  =  how  many  miles  ? 

18.  A  silver  dollar  which  weighs  412.5  grains,  contains 
41.25  grains  of  copper.  How  much  pure  silver  does  it  con- 
tain ?  371.25  grains. 

19.  A  lady  used  19  yards  of  silk  in  making  dresses  for  her 
two  daughters,  using  9.875  yards  for  one  dress.  How  much 
silk  did  the  other  dress  contain  ?  9.125  yards. 

20.  48.2175  acres  —  39.5  acres  are  how  many  acres? 

21.  At  a  certain  point  near  Sandy  Hook,  N.  J.,  the  water 
at  low  tide  is  5.649  feet  deep,  and  at  high  tide  it  is  11.249  feet 
deep.     How  much  does  the  tide  rise  and  fall  at  that  point  ? 

22.  The  walls  of  a  school  room  measure  120  square  yards, 
but  the  openings  (windows  and  doors)  measure  17.75  square 
yards.    How  many  yards  for  plastering  are  there  on  the  walls  ? 

23.  The  ceiling  of  the  same  room  measures  51.5  square 
yards.  How  many  yards  less  of  plastering  in  the  ceiling  than 
in  the  walls  ?  50.75. 

(34)  (25)  (26) 

87.006  1.03045  20000.2875 

9.84  .0009  482.52006- 

27.  From  eight  hundred  sixty  and  four  hundredths  take 
nmeteen  and  nine  thousand  eighty-four  hundred-thousandths. 

Bemainder^  840.94916. 

28.  A  cubic  foot  of  gold  weighs  1203.625  pounds,  and  a 
cubic  foot  of  iron  450.4375  pounds.  How  much  more  does 
the  gold  weigh  than  the  iron  ?  753.1875  pounds. 

29.  A  vessel  sailed  from  Boston  for  Havana  with'  a  cargo 
of  438.275  tons  of  ice,  but  156.895  tons  melted  on  the  voyp- 
How  much  of  the  ice  reached  Havana  ?  281.38  ' 

80.  How  much  more  ice  reached  Havana  than  rr 
voyage  ? 


108 


DECIMALS. 


SECTION   IV. 

MUZ  TI^ZICA.  TIOJV. 


.3 
.9 


.03 
_3 

.09 


.003 
3 

.009 


ca.se    I, 
One  Factor  an  Integer. 

136t  3  times  3  apples  are  9  apples,  3  times  3  ones 
are  9  ones,  and  3  times  3  tens  are  9  tens.  So,  also,  3 
times  3  tenths  are  9  tenths,  3  times 
3  hundredths  are  9  hundredths,  and 
and  3  times  3  thousandths  are  9 
thousandths.     (See  52,  V.) 

Ex.  1.  Multiply  43.21  by  4. 

Explanation. — We  write  the  factors  and  mul- 
tiply as  in  integers.  Since  4  times  1  hundredth 
are  4  hundredths,  and  4  times  2  tenths  are  8 
tenths,  the  4  in  the  product  is  hundredths, 
and  the  8  is  tenths.  We  must  therefore  place 
the  decimal  point  before  the  8. 

Ex.  2.  Multiply  .12815  by  7. 

Explanation.  —  Since  the  multiphcand  is 
hundred-thousandths,  7  times  this  multipli- 
cand must  also  be  hundred-thousandths  ;  and 
since  hundred-thousandths  are  expressed  by 
five  decimal  figures,  (see  123),  there  must  be 
five  decimal  figures  in  this  product.     Hence, 

When  the  multiplier  is  an  integer ,  there  are  as  many 
decimal  figures  in  the  product  as  in  the  multiplicand. 

PR  OBIjEMS. 

1.  In  one  rod  there  are  16.5  feet.     How  many  feet  across  a 
street  that  is  3  rods  wide  ?  Ji9.5. 

2.  How  many  acres  in  8  fields,  each  containing  12.84  acres  ? 


172.84 


SOLUTION. 

.12815 

7 

.89705 


MULTIPLICATION.  109 

3.  How  many  bushels  of  oats  are  there  in  4  bins,   solution  of 

i     •     •  .^/^  «^   1         1      1      ,.  PROBLEM  3. 

each  containing  30.75  bushels  ?  ^ 

4.  How  many  miles  will  a  man  travel  in  8  days,  4 
if  he  travels  44.635  miles  each  day  ?                357.           co^ 

5.  K  .085  of  a  pound  of  butter  be  made  from  one 

quart  of  milk,  how  much  butter  can  be  made  from  9  quarts  ? 

.765  of  a  -pound. 

6.  A  farmer  put  8  loads  of  hay  in  a  stack,  and  each  load 
weighed  .9375  of  a  ton.    How  many  tons  of  hay  in  the  stack  ? 

(7)                          (8)  (9)  (10) 
210.735                 634.04                 19.125                 250.375 
9                 6                          8  6 

11.  What  is  the  product  of  16.24  multiplied  by   solution  of 

14  ^  PKOBLEM  11. 

12.  If  a  man  builds  2.5  rods  of  stone-wall  in  one  \^ 
day,  how  many  rods  can  he  build  in  18  days  ?    Ji5.        tjtt 

13.  If  a  mason  can  plaster  145.75  square  yards      ig24 
of  wall  in  a  day,  how  many  yards  can  he  plaster  in     ^^„  „^ 
26  days?  S789.5. 

14.  A  merchant  bought  39  pieces  of  sheeting,  each  piece 
containing  40.25  yards.     How  many  yards  did  he  buy  ? 

15.  What  is  the  weight  of  47  reairfs  of  printing-paper,  each 
ream  weighing  38.125  pounds  ?  1791.875  'pounds. 

16.  How  many  rods  of  ditch  will  a  laborer  dig  in  64  days, 
if  he  digs  7.38  rods  each  day  ?  1^72.32. 

17.  A  jeweler  made  85  finger  rings,  using  .0225  of  an  ounce 
of  gold  for  each  ring.     How  much  gold  did  he  use  ? 

1.9125  ounces. 

18.  A  farmer  sheared  113  sheep,  and  the  fleeces  averaged 
4.0625  pounds.     What  was  the  amount  of  his  wool  clip  ? 

459.0625  pounds. 

19.  If  a  gardener  can  raise  405  bushels  of  onions  on  one  acre 
of  ground,  how  many  bushels  can  he  raise  on  3.276  acres  ? 

20.  How  much  is  7  million  times  7  millionths  ?  49. 

21.  If  94.3  tons  of  iron  are  required  for  one  mile  of  railroad, 
how  many  tons  will  be  required  for  a  road  164.35  miles  long  ? 


110 


DECIMALS, 


— U..j....;....i — i — j. — ji — ; — I..... 


CASE    II. 
Each  Factor  a  Decimal  or  a  Mixed  Number. 

137.  If  from  a  board  1  foot 
square  a  part  be  taken,  7  tenths 
of  a  foot  long  and  5  tenths  of  a 
foot  wide,  the  area  of  the  part 
may  be  found  by  multiplying  its 
length  by  its  breadth.     By  the 
diagram  it  will  be  seen  that  the 
square  foot  contains  10  times  10, 
or  100  small  squares  ;  and  hence, 
each  of  these  small  squares  is  1  hundredth  of  the  whole 
board.     The  part  taken  contains  5  times  7,  or 
35  of  these   small   squares,  or   35   hundredths         '7 
of  the  whole  board.     Hence,  7  tenths  multiphed       _:r 
by  5  tenths  must  produce  35  hundredths.  .35 

138i   The  product  of  tenths  multiplied  by  tenths  is  hun- 
dredths. 

Ex.  Multiply  24.3  by  .3. 

Explanation. — ^We  first  multiply  as  in  in- 
tegers. Then,  since  the  product  of  tenths 
multiplied  by  tenths  is  hundredths,  and  since 
hundredths  are  expressed  by  two  decimal 
figures,  we  place  the  decimal  point  before  the 
2  in  the  product. 

PBOB  LUJMS. 

23.  Multiply  .8  by  .3.  .         Product,  .U, 

23.  What  is  the  product  of  .9  multiplied  by  .7  ? 

24.  What  is  the  product  of  16.7  multipUed  by  .5  ?     8.35. 

25.  Multiply  38.3  by  8.  Product,  30.64. 

26.  If  a  man  can  chop  2.5  cords  of  wood  in  a  day,  how 
many  cords  can  he  chop  in  .7  of  a  day  ?  1.75. 

27.  How  many  square  feet  are  there  in  a  board  16.5  feet 
long  and  .9  of  a  foot  wide  ?  14.85. 


SOLTTTIOIf. 

24.3 
.3 

7.29 


MULTIPLICATION, 


111 


139.  If  from  a  cubic 
foot  a  part  be  taken, 
7  tenths  of  a  foot  long, 
4  tenths  of  a  foot  wide, 
and  3  tenths  of  a  foot 
thick,  the  capacity  of  the 
part  may  be  found  by 
multiplying  its  length, 
width,    and    thickness 
together.     By  the  first 
figure  it  will  be  seen 
that   the   whole    cubic 
foot  contains  10  times 
10  times  10,   or  1,000 
small  cubes  ;   and  hence  each  small 
cube  is  1  thousandth  of  the  cubic 
foot.     By  the  second  figure  it  will 
be  seen  that  the  jpart  taken  contains 
.7  times  4  times  3,  or  84  small  cubes, 
which   are   84   thousandths   of   the 
cubic  foot.     Hence,  the  product  of 
7  tenths,  4  tenths,  and  3  tenths  is  84 
thousandths. 

To  obtain  the  product  of  three  factors,  we 
multiply  the  product  of  the  first  two  by  the 
third.  The  product  of  7  tenths  and  4 -tenths  is 
28  hundredths.  Then,  28  hundredths  multi- 
plied by  3  tenths  must  produce  84  thousandths. 


.7 
^4 

.28 
.3 


.084 


140.  The  product  of  hundredths  multiplied  by  tenths 
is  thousandths. 

When  tenths  are  multiplied  by  tenths,  there  is  1 
decimal  figure  in  each  factor  and  2  iu  the  product. 
When  hundredths  are  multiphed  by  tenths,  there  are  2 
decimal  figures  in  one  factor,  1  in  the  other,  and  3 


112  DECIMALS. 

in  the  product.    In  each  case  there  are  as  many  deci- 
mal figures  in  the  product  as  in  both  factors. 

141.   The  product  must  always  contain  as        solution. 
many  decimal  figures  as  both  factors.  44.76 

Ex.  Multiply  44.76  by  8.23.  — '— 

Explanation. — Since  there  are  four  deci-  oqko 

mal  figures  in  both  factors,  there  must  be  35808 

four  decimal  figures  in  the  product.  qfifi  q74.« 

142.    ^ule  for  MuUipUcation  of  decimals, 

I.   Write  the  numbers,  and  multiply  as  in  integers. 
n.  Place  the  decimal  point  in  the  product  so  that  it  shall 
contain  as  many  decimal  figures  as  both  factors. 

m  OBLEMS. 

28.  What  is  the  product  of  .43  multiplied  by  .4  ?       .172. 

29.  Multiply  .84  by  .6.  Product,,  .504. 

30.  If  one  ton  of  iron  ore  yields  .685  of  a  ton  of  iron,  how 
many  tons  of  iron  will  893.056  tons  of  ore  yield  ?      611.7 Jf336. 

31.  How  many  tons  of  broom-corn  can  be  raised  from  .85 
of  an  acre  of  land,  if  1.8764  tons  can  be  raised  from  one 
acre  ?  l.BOJfOJ^. 

32.  How  many  gallons  of  linseed-oil  can  be  obtained  from 
249.5  bushels  of  flaxseed,  if  3.15  gallons  of  oil  can  be  obtained 
fi-om  one  bushel  of  seed  ?  785.925. 

33.  If  3.75  gallons  of  cider  can  be  made  from  one  bushel  of 
apples,  how  much  cider,  can  be  made  from  38.5  bushels  ? 

UJf.375  gallons. 

34.  If  one  yard  of  cassimere  can  be  made  from  1.625  pounds 
of  wool,  how  many  pounds  of  wool  will  be  required  for  54.25 
yards  ?  88.15625. 

35.  In  one  square  rod  there  are  272.25  square  feet.  How 
many  square  feet  in  108.4  square  rods  ?  29511.9. 

36.  If  22.73  pounds  of  starch  be  made  from  one  bushel  of 
com,  how  many  pounds  can  be  made  from  83.25  bushels  ? 


DIVISION.  113 

37.  Multiply  .0854  by  0.33. 

Since  there  are  seven  decimal  figures  in  both  solution  of 

factors,  and  in  the  product  but  five,  we  must  pre-  ^g^^ 

fix  two  ciphers  to  the  product,  and  place  the  *  Qg^ 

decimal  point  before  the  first  one.  v^ 

38.  Multiply  .084  by  .07.        Product,  .00588.  2552 

39.  What  is  the  product  of  .00393  multiplied  0027328 
by  .006?                                                      .00002358. 

40.  What  is  the  product  of  .057  and  .00049  ?     .00002793. 

41.  Multiply  .06052  by  .066.  Product, 


SECTION    V. 

^irisiojsr. 


CA.SE     I 
The  Divisor  an  Integer. 

143.  One  fourth,  of  8  apples  is  2  apples,  one  fourth 
of  8  feet  is  2  feet,  one  fourth  of  8  ones  is  2  ones,  and 
one  fourth  of  8  tens  is 

is  2  tens.    So  also  one      ^  [4        ^08  [4        ^008  [4 
fourth  of  8  tenths  is  2        .2  .02  .002 

tenths,  one  fourth  of  8 

hundredths  is  2   hundredths,  one  fourth  of  8  thou- 
sandths is  2  thousandths,  etc. 

Ex.  1.  Divide  .9275  by  7. 
Explanation. — ^We  write    the    numbers        solution. 
and  divide   as  in   integers.       Since   one      j??I?  I  ^ 
seventh  of  9  tenths  is  1  tenth  with  a  re-       .1325 
mainder,   the    first    quotient  figure   1   is 
tenths,  and  the  decimal  point  must  therefore  be  placed 
before  it. 


114  DECIMALS. 

Ex.  2.  Divide  1047.15  by  45. 

Explanation.  —  Di-      partial  solutiox, 
viding  1047  by  45  we      10^7.15  |  45 
obtain  a  quotient  of  23       -^-      [  23 
and  a  remainder  of  12,        147 
as  shown  in  the  Par-        1^^ 


SOLTJTIOX. 

1047.15  I  45 

90 


147 
135 


[  23.27 


tial   Solution.     Since         12  121 

the  three  of  the  quo-  90 

tient  is  ones,  we  must  place  the  deci-  325 

mal  point  after  it,  before  writing  the  315 

next  quotient  figure.     We  then  con- 
tinue the  division  until  all  the  figures  of  the  dividend 
have  been  used. 

144.  When  a  decimal  or  a  mixed  number  is  divided 
by  an  integer,  there  mill  be  as  many  decimal  figures  in 
the  quotient  as  in  the  dividend. 

PMOBJLEMS. 

1.  A  physician  fed  .675  of  a  ton  of  hay  to  his  horse  in  5 
weeks.     How  much  hay  did  he  feed  each  week  ?  .135  of  a  ton. 

3.  If  .804  of  a  ton  of  coal  will  last  a  family  6  weeks,  how 
much  do  they  bum  in  a  week  ?  .13 J},  of  a  tan. 

3.  A  bell-founder  cast  8  bells  of  equal  size,  and  they  weighed 
together  .984  of  a  ton.     What  was  the  weight  of  each  bell  ? 

.123  of  a  ton. 

4.  A  lady  put  16.24  pounds  of  grapes  into  8  fruit  cans. 
How  many  pounds  did  she  put  into  each  can  ?  2.03. 

5.  A  father  divided  217.5  acres  of  land  equally  among  his 
5  sons.     How  many  acres  did  each  son  receive  ?  Jf3.5. 

6.  If  2.1875  barrels  of  flour  will  last  a  family  7  months,  how 
much  flour  will  they  use  in  one  month  ?  .3125  of  larrel. 

145.  Ex.1.  Divide  4  by  8. 

Explanation. — Since  8  is  not  contained  in  4.  n  I  « 

4,  we  annex  a  decimal  cipher  to  the  four  ones  — ^ 
before  dividing.     (See  127.)  -^ 


DIVISION.  115 

Ex.  2.  How  many  times  is  32  contained  in  164  ? 
Explanation. — The    quotient   of   164  solution. 

divided  by  32  is  5  with  a  remainder  of      1^^ 

4,  and  since  the  quotient  figure  5  is [  5.125 

ones,  we  place  the  decimal  point  at  the  4000 

right  of  it.     We  then  annex   decimal  ^^ 

ciphers  to  the  remainder  4  ones  (see  80 

127),   and  continue  the    division  until  ^^ 

there  is  no  remainder.  160 

160 

PROBLEMS.  

7.  A  silver-ware  manufacturer  made  25  sets  of  table-spoons 
which  weighed  331  ounces.     How  much  did  one  set  weigh  ? 

9.2Ji,  ounces. 

8.  An  Iowa  farmer  raised  3045  bushels  of  corn  from  56 
acres.     How  much  was  the  yield  to  the  acre  ?     54.375  Imshds. 

9.  If  a  man  can  chop  44.635  cords  of  wood  in  31  days,  how 
many  cords  can  he  chop  in  one  day  ?  2.125. 

146.  Ex.  What  is  the  quotient  of  .099  di\ided  by  36  ? 

Explanation. — Since  one  thirty-sixth  solution 

of  99  thousandths  is  2  thousandths 
with  a  remainder,  we  write  the  2  m 
the  quotient  as  thousandths  by  pre- 
fixing two  decimal  ciphers,  and  then 

continue  the  division  until  there  is  no 

180 
remainder.  -  °J; 

PMOBLEMS.  

10.  The  dividend  is  .897  and  the  divisor  39.  What  is  the 
quotient  ?  .023. 

11.  If  .7505  of  an  ounce  of  gold-leaf  will  cover  79  square 
feet,  how  much  gold-leaf  will  be  required  to  gild  one  square 
foot  ?  .0095  of  an  ounce. 

13.  A  manufacturer  put  up  33  gallons  of  lemon  extract  in 
736  bottles.     How  much  did  eacl?  bottle  contain  ? 


.099 

36 

V2 

270 
252 

.  .00275 

116 


DECIMALS. 


13.  A  dairyman  made  7.2  pounds  of  butter  from  128  quarts 
of  milk.     How  much  butter  was  that  from  one  quart  of  milk  ? 

14.  Divide  .7  by  112.  Quotient,  .00625. 

15.  If  one  bushel  of  Onondaga  salt  can  be  made  from  35 
gallons  of  brine,  how  many  bushels  can  be  made  from  618.625 
gallons  ?  17.675. 

C^SE     II. 

The  Divisor 

147.  The  quo- 
tient of  15  divid- 
ed by  5  is  3.  If 
we  multiply  both 
dividend  and  di- 
visor by  4,  and 
divide  the  new 
dividend  by  the 
new  divisor,  the  quotient  will  be  3,  the  same  as  before. 
If  we  multiply  both  terms  by  23  and  divide,  we  obtain 
the  same  quotient. 

148«  ^  the  dividend  and  divisor  are  both  multiplied  by 
the  same  number,  the  quotient  remains  unchanged. 

Ex.  1.  Divide  91  by  .7.  soltttion. 

_,  _,_  _  Dividend.    91     I  .7  Divisor. 

Explanation. — ^We  mul-  iq    iq 

tiply  both  terms  by   10,  -—    — 

to    make    the    new  divisor    New  Dividend.  910   |  7   New  Divisor. 


a  Decimal  or  a  Mixed  Number. 

15   [5    Divisor. 

15    Dividend. 

3   Quotient. 

23 

45            5  Divisor. 

15         5  Divisor. 

30        23 

4      4 

345 

115NewDiv 

60 

20  New  Divisor. 

345 

3  Quotieni 

60 

3  Quotient. 

a  whole  number,  and  di- 
vide as  in  integers. 

Ex.  2.  Divide  32.5  by  .13. 

Explanation.  —  We  multi- 
ply both  terms  by  100,  to 
make  the  new  divisor  a  whole 
number,  and  divide  as  in  in- 
tegers. 


130   Quotient. 

SOLUTION. 

32.5       [  .13       Divisor. 

100      100 


3250 
26 


65 
65 


13    New  Divisor. 
250  Quotient. 


DIVISION. 


117 


Ex.  3.  Divide  6.95835 
by  .423. 

Explanation ^We  mul- 
tiply both  terms  by  1,000 
by  removing  the  deci- 
mal point  three  places 
to  the  right,  and  divide 
as  in  Case  I. 

Ex.  4.  Divide  .27  by  4.32. 

Explanation.  —  We  multi- 
ply both  terms  by  100  by 
removing  the  decimal  point 
two  places  to  the  right,  and 
divide  as  in  Case  I. 


80LTTTI0N. 

6.95835  [  .423  Divisor. 

6958.35 
423 


2728 
2538 
1903 
1692 


423    New  Divisor. 
•   16.45  Quotient. 


2115 
2115 


.27        [     4.32   Divisor. 
27.00  I      432   New  Divisor. 

2592  [   -^ 


.0625  Quotient. 


1080 
864 


From  these  examples  we  2160 

see  that  when    the    divisor  2160 

contains  one  decimal  figure, 

we  multiply  both  terms  by  10  ;  when  it  contains  two 
decimal  figures,  by  100 ;  when  three  decimal  figures,  by 
1,000,  and  so  on.     That  is, 

149i  Before  dividing,  both  terms  must  be  multiplied  by 
a  number  composed  of  1  with  as  many  ciphers  annexed  as 
the  divisor  contains  decimal  figures. 

16.  If  a  family  use  .75  of  a  barrel  of  flour  each  month,  how 
long  will  9  barrels  last  them  ?  12  months. 

17.  A  farmer  carried  15  bushels  of  wheat  to  mill,  and  re- 
ceived one  sack  of  flour  for  every  1.25  bushels.  How  many 
sacks  of  flour  did  he  get  ?  12. 

18.  If  2.125  yards  of  linen  can  be  made  from  .85  of  a 
pound  of  flax,  how  much  flax  will  be  required  for  one  yar^  of 
linen  ?  .^ofa  pouTid. 


118 


DECIMALS, 


19.  If  a  woman  can  weave  6.35  yards  of  rag  carpet  in  one 
day,  how  long  will  it  take  her  to  weave  40  yards  ?      64  days. 

20.  At  how  many  loads  can  a  teamster  draw  780.7  cubic 
yards  of  gravel,  if  he  draws  .925  of  a  yard  at  each  load?  84J^. 

21.  A  druggist  put  up  .875  of  gallon  of  sweet-oil  in  bottles, 
each  containing  .0625  of  a  gallon.  How  many  bottles  did  he 
fill?  U. 

22.  A  merchant  tailor  sold  21  yards  of  silk,  in  vest  patterns 
of  .875  of  a  yard  each.     How  many  patterns  did  he  sell  ?    2Jf. 

23.  If  a  farm  hand  can  plow  13.5  acres  of  land  in  one  week, 
how  long  will  it  take  him  to  plow  47.25  acres  ?      3.5  weeks. 

24.  What  is  the  length  of  a  lane  which  contains  21.12  square 
rods  and  is  .8  of  a  rod  wide  ?  26.J!i,  rods. 

25.  What  is  the  quotient  of  32.625  divided  by  43.5  ?       .75. 

26.  If  a  miller  makes  one  barrel  of  flour  from  4.5  bushels 
of  wheat,  how  many  barrels  will  he  make  from  49.5  bushels  ? 

27.  A  lady  put  up  58  quarts  of  strawberries  in  cans,  putting 
1.8125  quarts  in  each.     How  many  cans  did  she  fill  ?        82. 

28.  If  one  suit  of  clothes  can  be  made  from  5.75  yards  of 
cloth,  how  many  suits  can  be  made  from  109.25  yards  ?     19. 

150.    (Rule  for  division  of  S)eci77iats* 
I.  When  the  divisor  is  an  integer. 

1.  If  necessary,  annex  decimal  ciphers  to  the  dividend 
till  the  figures  of  the  dividend  will  contain  the  divisor, 

2.  Divide  as  in  whole  numbers. 

3.  Place  the  decimal  point  in  the  quotient  so  that  it  shall 
contain  as  many  decimal  figures  as  the  dividend. 

II.  When  the  divisor  is  a  decimal  or  a  mixed 
number. 

1.  Omit  the  decimal  point  from  the  divisor,  and  remove 
the  decimal  point  in  the  dividend  as  many  places  to  the  right 
as  the  original  divisor  contains  decimal  figures. 

2.  Divide  and  place  the  decimal  point  in  the  quotient  as 
before. 


DIVISION 


119 


I^BOBLJEMS. 

29.  Divide  7  by  43.75.  OvMimt,  .16, 

30.  Divide  525  by  9.375.  Quotient^  56. 

31.  If  .625  of  a  yard  of  cloth  be  made  from  one  pound  of 
wool,  how  many  pounds  of  wool  will  be  required  for  12 
yards  of  cloth?  19.2. 

32.  A  farmer  cut  639  cords  of  wood  from  11.25  acres  of 
woodland.     How  many  cords  was  that  to  the  acre  ?       56.8. 

33.  In  how  many  hours  can  you  empty  a  cistern  that  con- 
tains 204  barrels  of  wafer,  if  you  pump  out  18.75  barrels  each 
hour  ? 

34.  If  a  steamboat  runs  156.25  miles  in  a  day,  in  how  long 
a  time  will  it  run  80  miles  ?  .512  of  a  day. 

35.  How  much  land  will  be  required  to  raise  21  bushels  of 
corn,  if  the  yield  is  at  the  rate  of  65.625  bushels  per  acre  ? 

36.  If  112.59  pounds  of  maple  sugar  are  made  from  625.5 
gallons  of  sap,  how  much  sugar  can  be  made  from  one  gallon 
of  sap  ?  .18  of  a  pound. 

37.  The  dividend  is  28  and  the  divisor  .64.  What  is  the 
quotient  ?  Jf3.75. 

38.  What  is  the  quotient  of  85  -^  .272  ?  312.5. 

39.  Divide  267.66  by  11.896.  Quotient,  22.5. 

40.  A  perfumer  put  up  73  gallons  of  cologne  in  bottles, 
putting  .0625  of  a  gallon  into  each.  How  many  bottles  did 
he  fill  ?  1168. 

41.  How  many  cars  will  be  required  to  transport  80410.5 
tons  of  freight,  allowing  each  car  to  carry  6.2625  tons  ?  12840. 

42.  How  many  blocks  of  marble  each  weighing  .9376  of  a 
ton  will  together  weigh  15.9392  tons  ?  17. 

43.  If  one  gallon  of  alcohol  can  be  made  from  .38  of  a 
bushel  of  corn,  how  many  gallons  can  be  made  from  15.39 
bushels  ?  40.5. 

4:4:.  In  a  fence  .9  of  a  mile  long  how  many  lengths  are  there, 
each  length  being  .00225  of  a  mile  long?  4OO. 

45.  If  .0196  of  a  cord  of  wood  will  make  one  bushel  of  char- 
coal, how  many  bushels  will  .833  of  a  cord  make  ?        4^.5. 


120  DECIMALS. 

CASE    III. 
True  Remainders. 

151.  Ex.  How  many  barrels  each  holding  3.5  bushels 
can  be  filled  from  237  bushels  of  apples,  and  how  many 
apples  will  be  left  ? 

Explanation.  —  Since  the  divisor  solution. 

contains  one  decimal  figure,  we  mul-  ^^ '       ^-^ 

tiply  both  terms  by  10  before  com-  ^370  I  35 

mencing  to  divide.    But  since  2,370,  ^-^^    I  67  barrels. 
the  dividend  used,  is  10  times  as        270 
great  as  the  given    dividend,   the        ^^ 
remainder  25,  which  is  a  part  of  25  Remainder, 

this  2,370,  is  ten  times  as  ffreat  as         fT^^  .  , 

.  mi  o  -(5. 0  True  remainder. 

the  true  remainder.     Thereiore,  to 
find  the  true  remainder,  we  divide  the  25  by  10,  which 
we  do  by  placing  a  decimal  point  before  the  5.    Hence, 
67  barrels  and  2.5  bushels  over  is  the  result  required. 

152.  When  the  quotient  is  an  integer,  the  true  remainder 
must  always  contain  as  many  decimal  figures  as  there  are 
in  the  given  term  having  the  more  decimal  figures. 

PROBJOEMS, 

46.  A  farmer  has  494  gallons  of  cider.  How  many  barrels 
can  he  fill  putting  31.5  gallons  into  each  ? 

15 f  and  heme  21.5  gallons  left. 

47.  If  9.5  tons  of  freight  make  1  car  load,  how  many  car 
loads  are  124.3  tons  ?  .1!  of  a  ton  more  than  13  car  loads. 

48.  A  farmer  who  has  134  bushels  of  wheat,  wishes  to  ex- 
change it  for  sheep.  If  he  gives  3.5  bushels  for  1  sheep,  how 
many  sheep  can  he  buy  ?    55,  and  haw  1.5  lushels  of  wheat  left. 

49.  Into  how  many  building  lots  each  containing  1.35  acres 
can  I  divide  9.5  acres  ?    Into  7,  and  .75  of  an  acre  remaining. 

50.  A  quartermaster  has  834.35  pounds  of  coffee.  If  he  dis- 
tributes 71.35  pounds  to  his  regiment  daily,  how  many  days' 
rations  of  coffee  has  he,  and  how  much  over  ? 

11  days'*  rations^  and  50.5  pounds  over. 


UNITED     STATES     MONEY.  121 

SECTION    VI. 
ujsriT^:^  STATBS  MOj\rBr. 

153*  United  States  Money  —  also  called  Federal 
Money — consists  of  dollars,  cents,  and  mills. 

10  mills  are  1  cent.         I        1  dollar  is  100  cents. 
100  cents  are  1  dollar.      I        1  cent  is  10  mills. 

154.  Since  100  cents  are  1  dollar,  1  cent  is  1  hun- 
dredth of  a  dollar,  and  is  written  $.01.     And 

155.  Since  10  mills  are  1  cent,  1  mill  is  1  tenth  of  a 
cent  or  1  thousandth  of  a  dollar,  and  is  written  $.001. 

The  divisions  of  a  dollar  into  cents  and  mills  corres- 
pond to  the  decimal  divisions  of  a  doUar  into  hun- 
dredths and  thousandths.     Hence, 

156.  Gents  may  always  he  written  as  hundredths,  and 
mills  as  thousandths,  of  a  dollar. 

85  cents  are  written  $.35. 

8  cents  are  written  $.08.        |      6  mills  are  written  $.006. 

10  cents  5  mills  are  written  $.105. 

7  dollars  93  cents  are  written  $7.93. 

5  dollars  56  cents  8  mills  are  written  $5,568. 

20  dollars  30  cents  1  mill  are  written  $20,201. 

EXEIt  CIS  ES. 

1.  Eead  $.15,  $.60,  $318.75,  $14.06,  $5.94,  $8.01. 

2.  Eead  $.255,  $.004,  $300,567,  $12,108,  $575.10,  $900.25. 

3.  Write  37  cents,  80  cents,  6  cents. 

4.  Write  13  dollars  4  cents,  75  dollars  50  cents. 

5.  Write  8  mills,  15  cents  6  mills. 

6.  Write  83  dollars  12  cents  5  mills. 

7.  Write  400  dollars  8  cents  1  mill. 

157.  Decimal  parts  of  a  dollar  less  than  mills  or  thou- 
sandths  are  read  as  decimals  of  a  mill. 

$.0006  is  6  tenths  of  a  mill ;  $.0085  is  8  and  5  tenths  mills. 
$.2943  is  29  cents  4  and  3  tenths  mills. 
$15.65425  is  15  dollars  65  cents  i  and  25  hundredths  mills. 
K 


122 


DECIMALS 


oom:i>ut.a.xions  of  xj.  s.  :m:one;y 

158.  Ex.  1.  What  is  the  sum  of  $108.50, 
$10,875,  and  $.458? 

Explanation.  —  We  write  the  numbers 
with  dollars  under  dollars,  cents  under 
cents,  and  mills  under  mills  ;  and  then  add 
the  parts,  and  place  the  decimal  point  in  the 
sum,  as  in  Addition  of  Decimals. 

Ex.  2.  From  $45.25  subtract  $17,625. 
Explanation. — We  write  the  numbers  with 
dollars  under  dollars,  cents  under  cents,  and 
mills  under  mills  ;  and  then  subtract,  and 
place  the  decimal  point  in  the  remainder, 
as  in  Subtraction  of  Decimals. 


SOLUTION. 

$108.50 

10.875 
.458 

1119.833 


SOLUTION. 

145.250 
17.625 

$27,625 


Ex.  3.  Multiply  $8,126  by  2.7. 
Explanation.  —  We  write   the  multipHer 
under  the  multipHcand  ;  and  then  multiply, 
and  place  the  decimal  point  in  the  product, 
as  in  Multiphcation  of  Decimals.  i 


SOLUTION. 

$8,125 

2.7 


56875 
16250 

$21.9375 


Ex.  4.  Divide  $436.72  by  53. 

Explanation. — We  write  the  divisor 
at  the  right  of  the  dividend  ;  and  then 
divide,  and  place  the  decimal  point  in 
the  quotient,  as  in  Division  of  De- 
cimals. 

Ex.  5.  How  many  times  are  127.50 
contained  in  $1168.75? 

159.  In  business,  when  the  mills 
in  any  final  result  are  5  or  more, 
they  are  regarded  as  1  cent,  and 
when  less  than  5,  they  are  rejected. 


$436.72 
424 


127 
106 


53 

$8.24 


212 
212 

SOLUTION, 

$1168.75 
116875 
11000 

6875 
5500 


13750 
13750 


UNITED     STATES     MONEY.  123 

PJROB  IjEMS. 

Find  the  sum  of  the  several  amounts  of  money  in  the  first 
four  problems. 


(1) 

(3) 

(3) 

(4) 

$121.10 

$     7.28 

$.58 

$2000 

38.47 

241.09 

.145 

5.75 

92.86 

.42 

.0275 

48.01 

?82.79 

.96 

.5625 

.495 

810.04 

44.52 

.095 

859.17 

5.  One  day  a  toll-gate  keeper  received  $17.56,  and  the  next 
day  $28.25.     How  much  toll  did  he  receive  in  the  two  days  1 

6.  A  furniture  dealer  sold  a  wash-stand  for  $6.50,  a  bureau 
for  $11.63,  and  a  rocking-chair  for  $8.25.  For  how  much  did 
he  sell  all  of  them  ?  $26.38. 

7.  On  Saturday  night  a  laborer  paid  $1,625  for  flour,  $.85 
for  tea,  $.75  for  sugar,  and  $.375  for  butter.  How  much 
money  did  he  pay  out  ?  $3.60. 

(8)  (9)  (10)  (11) 

From      $250.35  $.104  $100,000  $1,000 

take  187.50  .087  5.875  .065 

12.  I  owe  $167.45.  If  I  pay  $94.50,  how  much  will  I 
then  owe  ?  $72.95. 

13.  A  lawyer  having  $256.56  in  the  bank,  drew  out  $98.75. 
How  much  money  had  he  left  in  the  bank  ?  $157.81. 

14.  One  week  a  laborer  earned  $12.50,  and  expended  $8.38. 
How  much  of  his  earnings  had  he  left  ?  $Ji,.12. 

(15)                  (16)                 (17)                 (18) 
Multiply   $194.17            $310.75            $50.44            $249.60 
by  8  36  7.5  ^ 

19.  How  much  will  7  bushels  of  wheat  cost  at  $1,125  a 
bushel  ?  -  $7,875. 

20.  If  an  ounce  of  indigo  costs  $.15,  how  much  will  9 
ounces  cost  ?  $1.35. 

21.  A  builder  bought  37  thousand  feet  of  pine  lumber  at 
$28.25  a  thousand.     How  much  did  it  cost  him  ?       $10^5.25. 


124  DECIMALS. 

22.  How  much  will  8.5  gallons  of  kerosene  cost  at  $.625  a 
gallon  ?  $5.3125. 

Find  the  quotient  in  problems  23,  24,  25,  and  26. 

(23)  (24)  (25)  (26) 

$51.75  1  9      $156.06  I  8^      $405.65  I  $21.35      $7.82  I  $.085 

27.  If  7  pounds  of  sugar  cost  $.91,  what  is  the  price  of  a 
pound  ?  $.13. 

28.  At  what  price  per  head  must  I  sell  105  sheep,  to  receive 
$564,375  for  them  ?  $5,375. 

29.  A  grocer  paid  $35  for  a  barrel  of  sugar  at  $.125  a  pound. 
How  many  pounds  did  he  buy  ?  280. 

30.  A  chair  maker  received  $172.50  for  chairs  at  $7.50  a 
set.     How  many  sets  did  he  sell  ?  23. 

31.  If  31.5  gallons  of  linseed-oil  cost  $59.0625,  what  is  the 
price  of  a  gallon  ?  $1,875. 

160«  ^ule  for  Computations   of  ZTnited  States 
JKoney, 

Write  the  numbers,  and  add,  subtract,  multiply,  divide, 
and  place  the  decimal  points  in  the  results,  as  in  Decimals. 

Pit  om^EMS. 

32.  A  man  bought  a  village  lot  for  $325,  and  after  paying 
$22.63  for  taxes,  he  sold  it  so  as  to  gain  $72.37.  For  how 
much  did  he  sell  it  ?  $420. 

33.  I  bought  a  hat  for  $4,875,  a  coat  for  $28,  and  a  pair  of 
boots  for  $7.50.     How  much  did  they  cost  me  ?       $40,375. 

34.  A  farmer  sold  a  jar  of  butter  to  a  merchant  for  $10.37, 
receiving  in  payment  groceries  to  the  amount  of  $4.63,  and  the 
balance  in  money  ?     How  much  money  did  he  receive  ?  $5.74. 

35.  How  much  will  125  pounds  of  nails  cost  at  $.06  a 
pound  ?  $7.50. 

36.  What  will  be  the  cost  of  .84  of  a  ton  of  plaster  at  $4.25 
a  ton  ?  $3.57. 


REVIEW     PROBLEMS.  125 

37.  At  $2.50  a  bushel,  how  many  plums  can  be  bought  for 
$1,875?  .75ofalushel. 

38.  How  much  will  29  rolls  of  wall-paper  cost  at  $.44  a 
roll  ?  $12.76. 

39.  How  much  muslin  can  be  bought  for  $24.36,  at  $.56  a 
yard  ?  JfS.B  yards. 

40.  I  paid  $.78  for  a  piece  of  fresh  beef  at  $.12  a  pound. 
How  much  did  it  weigh  ?  6.5  pounds. 

41.  A  mechanic  earned  $56.25  in  January,  $45.63  in  Feb- 
ruary, $67.50  in  March,  $65,875  in  April,  and  $75  in  May. 
How  much  did  he  earn  in  the  five  months  ?  $310,255. 

42.  Mr.  Stevens  bought  a  watch  for  $32.25,  and  sold  it  to 
Mr.  Adams  for  $27.75.  How  much  did  he  lose  by  the  trans- 
action ?  $^.50. 

43.  Mr.  Adams  afterward  sold  it  for  $30,625.  How  much 
did  he  gain  ?  $2,875. 

44.  One  season  a  nurseryman  sold  2840  young  apple-trees 
at  $.375  apiece.     How  much  did  he  receive  for  them  ?    $1065. 

45.  A  stationer  paid  $114  for  pocket-knives  at  $.95  apiece. 
How  many  did  he  buy  ?  120. 

46.  How  much  must  be  paid  for  transporting  .456  of  a  ton 
of  freight  from  New  York  to  Toledo  by  railroad,  at  $28.60  a 
ton  ?  $13.0416. 

47.  A  fruit  dealer  sold  686  baskets  of  peaches  for  $1543.50. 
What  was  the  price  per  basket  ?  $2.25. 


SECTION  VII. 

TliOSZBMS  IJ\r  ^£JClMjiZS, 

1.  A  merchant  deposited  $59.17  in  the  bank  on  Monday, 
$62.86  on  Tuesday,  $48.12  on  Wednesday,  $75.48  on  Thurs- 
day, $88.57  on  Friday,  and  $110.72  on  Saturday.  What  was 
the  amount  of  his  deposits  for  the  week  ?  $Jf.Jf,Jf..92. 

2.  What  vrill  be  the  cost  of  15,000  bushels  of  wheat  at 
$2.0625  a  bushel  ?  $30937.50. 


126  DECIMALS. 

3.  When  the  price  of  rice  is  $.0625  a  pound,  how  many- 
pounds  can  be  bought  for  $3.50  ?  56. 

4.  If  wheat  is  worth  $1.4375  per  bushel  in  Chicago,  and 
$2,125  in  New  York,  how  much  is  added  to  its  value  by 
transportation  ?  $.6875  per  bushel. 

5.  How  much  will  it  cost  to  build  18.4  rods  of  picket-fence 
at  $3,125  a  rod  ?  $57.50. 

6.  A  fruit  dealer  paid  $14.25  for  95  quarts  of  strawberries. 
How  much  did  he  pay  a  quart  for  them  ?  $.15. 

7.  I  cut  .912  of  a  ton  of  hay  from  my  door  yard,  and  the 
yield  was  at  the  rate  of  1.92  tons  to  the  acre.  How  much 
land  is  there  in  my  door  yard  ?  .^75  of  an  acre. 

8.  When  cranberries  are  worth  $5.00  a  bushel,  what  part  of 
a  bushel  can  be  bought  for  $.625  ?  .1^5. 

9.  A  shoemaker  paid  $385.40  for  sole-leather,  $216.94  for 
upper-leather,  $104.05  for  linings,  $24.28  for  thread  and  silk, 
and  $12.75  for  pegs.     How  much  did  he  pay  out  for  stock  ? 

10.  What  is  the  quotient  of  .016  -^  .512  ?  .03125. 

11.  When  the  dividend  is  .01  and  the  divisor  12.8,  what  is 
the  quotient  ?  .00078125, 

12.  Mr.  Butler  had^  a  silver  watch  worth  $18.75,  which  he 
exchanged  for  a  gold  watch  worth  $80,  paying  the  balance  in 
money.     How  much  did  he  pay  to  boot  ?  $61.25, 

13.  If  you  pay  $24  for  7.5  reams  of  letter-paper,  how  much 
does  it  cost  you  a  ream  ?  $3.20. 

14.  How  many  packages  each  containing  .875  of  a  pound  can 
be  filled  from  a  chest  which  contains  55  pounds  of  tea,  and 
how  much  tea  will  be  left  ?      62  packages  ;  .75  of  a  pound  left. 

15.  A  man  on  a  journey  paid  $32.17  for  railroad  fare,  $12.44 
for  steamboat  fare,  $37.25  for  hotel  bills,  and  $7.32  for  other 
expenses.    What  were  his  total  expenses  ?  $89.18. 

16.  How  much  hemlock  bark  at  $8.25  a  pord  will  pay  a 
biU  of  $57.75  for  boots  and  shoes  ?  '  7  cords. 

17.  At  $4,375  a  head,  how  much  will  1000  sheep  cost  ? 

18.  A  merchant's  sales  in  September  were  $2174.15,  in 
October  $1416.24,  in  November  $1765.93,  and  in  December 
$2443.76.     How  much  did  his  sales  average  per  month  ? 


KEVIEW     PROBLEMS.  127 

19.  A  man  who  owed  $250,  paid  at  one  time  $65.48,  at 
another  time  $47.81,  and  at  another  $93.37.  How  much  did 
he  then  owe  ?  $UM' 

20.  What  is  the  quotient  of  .315  -=-  .3125  ?  1.008. 

21.  If  my  expenses  for  five  consecutive  weeks  are  $12.25, 
$13.61,  $14.09,  $11.52,  and  $13.78,  how  much  are  my  average 
weekly  expenses  ?  $13.05. 

22.  A  fruit  dealer  paid  $15.45  for  oranges,  $20.34  for 
lemons,  $27.59  for  pine-apples,  and  $16.72  for  cocoa-nuts. 
How  much  did  the  fruit  cost  him  ?  $80.10. 

23.  If  I  buy  goods  to  the  amount  of  $8.45,  and  in  paying 
for  them  give  a  10-dollar  bill,  how  much  change  ought  I  to 
receive  ?  $1.55. 

24.  A  merchant  tailor  sold  a  piece  of  damaged  cloth  at  a 
loss  of  $19.88,  and  the  cloth  cost  him  $87.50.  For  how  much 
did  he  sell  it  ?  $67.62. 

55.  A  dealer  in  agricultural  implements  paid  $199.50  for 
plows,  at  $7,125  each.     How  many  plows  did  he  buy  ?    28. 

26.  What  will  be  the  cost  of  32.5  yards  of  tapestry  carpet 
at  $2.75  a  yard  ?  $89,375. 

27.  If  1  coat  can  be  made  from  3.125  yards  of  cloth,  h'ow 
many  can  be  made  from  52.5  yards  ? 

16,  with  a  remnant  of  2.5  yards. 

28.  A  spice  dealer  put  up  280  pounds  of  ground  cinnamon 
in  boxes,  each  holding  .25  of  a  pound.  How  many  boxes  did 
he  fill  ?  1120. 

29.  From  a  piece  of  cloth  containing  44.5  yards,  a  tailor 
made  as  many  suits  of  clothes,  each  containing  8.375  yards,  as 
he  could.     How  many  yards  were  left  in  the  piece  ?    2.625. 

30.  How  many  pounds  of  metal  will  it  take  to  cast  one 
church  bell  weighing  2,765  pounds,  and  7  factory  bells,  each 
weighing  325  pounds  ?  5,0J^0. 

31.  From  a  cask  which  contained  37.175  gallons  of  alcohol, 
a  druggist  drew,  at  different  times,  .125  of  a  gallon,  1.5  gal- 
lons, .25  of  a  gallon,  2.75  gallons,  .625  of  a  gallon,  .0625  of  a 
gallon,  and  .75  of  a  gallon.  How  many  gallons  were  then  left 
in  the  cask  ?  31.1125. 


128  DECIMALS. 

32.  How  much  delaine  at  $.5635  a  yard  can  he  bought  for 

$9  ?  16  yards. 

33.  A  butcher  paid  $58.60  for  an  ox,  and  after  killing  it,  he 
retailed  the  meat  for  $59.76,  sold  the  tallow  for  $4.18,  and 
the  hide  for  $7.88.     How  much  were  his  profits  ?       $  13.22. 

34.  A  provision  dealer  bought  pork  at  $.125  a  pound,  and 
sold  it  at  $.13.     How  much  did  he  gain  a  pound  ? 

35.  If  a  family  use  .785  of  a  barrel  of  flour  in  one  month, 
.825  of  a  barrel  the  next  month,  .73  of  a  barrel  the  third,  .8  of 
a  barrel  the  fourth,  and  .76  of  a  barrel  the  fifth,  what  is  the 
average  amount  used  per  month  ?  .78  of  a  Mrrd. 

36.  A  farmer  harvested  713.5  bushels  of  oats  from  a  field  of 

13  acres,  and  576.25  bushels  from  a  field  of  9  acres.  What  was 
the  average  yield  per  acre  of  the  two  fields  ?   58.625  dushels. 

37.  11  miles  of  a  certain  railroad  cost  $13875.30  per  mile, 

14  miles  cost  $15251.64  per  mile,  and  28  miles  cost  $14588.45 
per  mile.  What  was  the  length  of  the  road,  and  what  the 
average  cost  per  mile  ?  Cost  per  mile,  $1^615.62. 

38.  George  Wells  bought  5  yards  of  casimere  at  $1,875, 
1  yard  of  alpaca  for  $.875,  13  yards  of  calico  at  $.25,  and  14 
yards  of  muslin  at  $.35.     What  was  the  cost  of  the  whole  ? 

(See  Manual,  page  219-) 

^ouj,/^  ojf  f.  B.  ^hams  S^  (^a, 

S/a/tei<f  ^aic/e^   Mea!^,  (^  ^.  OS", /     .^0 

S  aual'/d  ^CaUcno^ai^  ^a<f,    ^^        ./S^ .-^S 

/<^ac/Me^, S.SO 

/ ^aic/eTt  Ma/e, ./jT 

/M^,   ^P.SS;     /S^iac/e,   ^/.SS, /O.  SO 

i>  ^oe^,  /^«^y//4; /^-^ 


'(/  tJaymen/, 


c^  3.    ^c/a^  f^o. 


CHAPTER   III. 
COMPOUND    NUMBERS. 

SECTION  I. 

J\rOTA.TIOJV  :dJV^    "E^D ZfC TIOJV. 

161*  We  find  the  amount  or  quantity  of  articles 
bought  and  sold,  by  measuring  or  ^Yeighing  them. 

Some  articles  are  sold  by  the  quart  or  gallon  ;  some 
by  the  peck  or  bushel ;  some  by  the  foot  or  yard  ;  some 
by  the  acre  ;  some  by  the  cord,  and  some  by  the  pound 
or  ton. 

162.  The  names  applied  to  particular  amounts  or 
quantities  of  articles  are  Denominations  ;  as  the  gallon, 
bushel,  foot,  mile,  pound,  and  dozen. 

163.  Numbers  applied  to  denominations  are  Denomi- 
nate Numbers  ;  as  4  yards,  9  bushels. 

164.  A  number  expressed  in  two  or  more  denomina- 
tions is  a  Compound  Number ;  as  4  hours  30  minutes, 
3  yards  2  feet  6  inches. 

A  denominate  number  may  be  an  integer,  as  3  bush- 
els ;  a  decimal,  as  .5  of  a  mile  ;  a  mixed  number,  as 
6.75  tons,  or  a  compound  number,  as  5  pounds  6  ounces. 

In  writing  compound  numbers,  the  denominations 
are  generally  abbreviated,  as  shown  in  the  tables. 

(See  Manual,  page  219.) 

165.  Those  denominations  in  a  compound  number 
which  express  the  greater  amount  are  Higher  Denomi- 
nations:  and 

L 


130 


COMPOUND     NUMBERS. 


166t  Those  which  express  the  less  amount  are  Lower 
Denominations.  Thus,  a  quart  is  a  higher  denomina- 
tion than  a  pint,  and  a  lower  denomination  than  a 
gallon. 

167.  Changing  numbers  from  one  denomination  to 
another  without  changing  their  value  is  Reduction. 

168.  Keducing  numbers  from  higher  to  lower  de- 
nominations is  Eeduction  Descending  ;  and 

169.  Keducing  Numbers  from  lower  to  higher  de- 
nominations is  Reduction  Ascending. 


170.  Table  I,— Liquid  Pleasure. 

The  denominations  gallons,  quarts,  pints,  and  gills 
constitute  Liquid  Measure.  They  are  used  in  measur- 
ing oil,  molasses,  wines,  milk,  and  other  liquids. 


4  gi.  (gills)  are  1  pt.  (pint). 

3  pt.  "    1  qt.  (quart). 

4  qt.  "    1  gal.  (gallon). 


1  gal.  is  4  qt. 
1  qt.  "  2  pt. 
1  pt.    "  4  gi. 


NOTATION     AND    REDUCTION.  131 

EXBJl  CISJES. 

1.  Read  5  gal.  3  qt.  1  pt.  1  gi. ;  14  gal.  3  qt.  1  pt. 

2.  Read  11  gal.  1  pi.  2  gi. ;  7  gal.  1  pt. ;  3  qt.  1  gi. 

3.  Write  fisre  gallons  one  quart  one  pint  tvvo  gills. 

4.  Write  seventeen  gallons  two  quarts  three  gills. 

REDXJCTIOIN"    DESCENDING^. 


SOLTTTION, 


171.  Ex.  1.  How  many  pints  are   equal 

to  3  gaUons  ?  ^  9^^- 

Explanation. — Since  3  gal.  are   3  times  — 

1  gal.,  and  1  gal.  is  4  qt.,  3  gal.  are  3  times  ^  ^^' 

4  qt.,  or  12  qt.  ;    and  since  12  qt.  are  12  — 

times  1  qt.,  and  1  qt.  is  2  pt.,  12  qt.  are  12  24^^. 
times  2  pt.,  or  24  pt.                            Hence,  3  gal.  = 


Ex.  2.  How  many  pints  are  equal  to  2  gal.  3  qt.  1  pt.? 

Explanation. — Since    2 

gal.   are  2  times  1   gal.,  solution. 

and  1  gal.  is  4  qt,  2  gal.  2  gal.  3  qt.  1  pt. 

are  2  times  4  qt.,  or  8  qt.,  - 

and  8  qt.  +  3  qt.  are  11  8  +  3  =  11  qt. 

qt.     Since  11  qt.  are  11  _2 

times  1  qt.,  and  1  qt.  is  2  22  +  1  =  23  pt. 
pt.,  11  qt.  are  11  times  2 

pt,  or  22  pt.,  and  22  pt.  +  Hence,  2  gal.  3  qt.  1  pt.  =  23  pt. 

1  pt.  are  23  pt. 

PMOBIjEMS. 

1.  Reduce  11  gallons  to  quarts.     (See  Ex.  1.)  J^  qt. 

2.  How  many  gills  are  there  in  4  quarts  ?  32. 

3.  A  wholesale  druggist  put  ten  gallons  of  sweet-oil  into 
bottles  which  held  1  gill  each.  How  many  bottles  did  he 
fill  ?  320. 

4.  How  many  pint  bottles  will  be  required  to  hold  7  gal- 
lons of  currant  wine  ?  56. 

5.  In  2  gal.  3.  qt.  1  pt.  3  gi.,  how  many  gills  ?  (See  Ex.  2.)  95. 


132 


COMPOUND     NUMBERS, 


6.  A  druggist  lias  11  gal.  2  qt.  of  alcohol.  How  long  will 
it  last  him,  if  he  sells  1  qt.  each  day  ?  JjB  days. 

7.  How  many  pint  bottles  will  4  gal.  3  qt.  1  pt.  of  catchup  fill  ? 

8.  A  grocer  bought  a  barrel  containing  31.5  gallons  of  vine- 
gar, which  he  sold  by  the  quart.  How  many  quarts  did  he 
sell  ?  126, 

9.  Reduce  5  gal.  3  qt.  1  pt.  3  gi.  to  gills. 


RKJDXJCTlOlSr    ASCENDING-. 

172.  Ex.  1.  How  many  gallons  are  48  pints  ? 

Explanation.  —   Since  solution. 

every  2  pt.  are  1  qt.,  and 
2  pt.  are  contained  in  48 
pt.  24  times,  48  pt.  are  24 
qt.  And  since  every  4  qt. 
are  1  gal.,  and  4  qt.  are 
contained  in  24  qt.  6 
times,  24  qt.  are  6  gal. 

Ex.  2.  How  many  gallons  in  79  pints  ? 

Explanation.  —  Since 
every  2  pt.  are  1  qt.,  and 
2  pt.  are  contained  in  79 
pt.  39  times  with  a  re- 
mainder of  1  pt.,  79  pt. 
are  39  qt.  1  pt.  And 
since  every  4  qt.  are  1  gal., 
and  4  qt.  are  contained 
in  39  qt.  9  times  with  a 
remainder  of  3  qt.,  39  qt. 
are  9  gal.  3  qt. 

In  the  first,  or  EuU  So- 
lution, we  have  written  all 
the  numbers  mentioned 
in  the  explanation,  both 


4^pL  ["Ipt. 
24  times. 

MqL  [4  qt 

6  times. 
24  qt.  =  6  gal 
Hence,  4S  pt  =  6 


FULL   SOLUTION. 

Idpt.  [2pt. 

39  times  and  1  pt.  rem. 
7dpt.  =  3dqt.  Ipt. 
ddqt.  [4  qt. 

9  times  and  3  qt.  rem. 
39  qt.  =  9  gal  3  qt. 

Hence,  79  pt.  =  9  gal  3  qt.  1  pt, 

COMMON  SOLUTION. 

Idpt.  [2pt. 
39  qt.  Ipt  [4  qt 
9  gal  3  qt 
Hence,  79  pt  =  9  gal  3  qt.  1  pt. 


NOTATION     AND     REDUCTION.  133 

abstract  and  concrete  ;  but  in  tlie  second,  or  Common 
Solution,  we  have  omitted  the  abstract  quotients,  and 
written  only  the  denominate  numbers. 
rnoBJj^EMS. 

10.  How  many  gallons  are  356  quarts  ?     (See  Ex.  1.) 

11.  How  many  gallons  of  cider  will  it  take  to  fill  104  pint 
bottles  ?  13. 

13.  Reduce  160  gills  to  gallons.  5  gal 

13.  Reduce  140  gills  to  higher  denominations.     (See  Ex.  3.) 

4  gal.  1  qt.  1  pt. 

14.  A  woman  buys  of  a  milkman  1  pt.  of  milk  a  day.  How 
much  does  she  buy  in  a  year  or  365  days  ?  JjS  gal.  2  qt.  1  pt. 

15.  In  one  week  a  grocer  sold  190  quart  cans  of  oysters. 
How  many  gallons  of  oysters  did  he  sell  ?  Ji7.5. 

16.  Reduce  655  gills  to  higher  denominations. 

17.  One  morning  a  farm  hand,  in  watering  cattle,  pumped 
879  strokes,  and  at  each  stroke  the  pump  discharged  1  pint 
of  water.     How  much  water  did  he  pump  ? 

109  gal.  3  qt.  1  pt. 

We  have  now  learned  that  gallons  are  reduced  to 
quarts,  quarts  to  pints,  and  pints  to  gills,  by  multiply- 
ing ;  and  that  gills  are  reduced  to  pints,  pints  to  quarts, 
and  quarts  to  gallons,  by  dividing. 

Since  the  reduction  of  gallons  to  gills  is  from  a  higher 
to  a  lower  denomination,  and  the  reduction  of  gills  to 
gallons  is  fi'om  a  lower  to  a  higher  denomination,  the 
explanations  already  given  are  sufficient  to  establish 
the  following 

173.    General  Principles  of  deduction, 

I.  A  denominate  number  is  reduced  to  lower  denominor 
tions  by  multiplication. 

n.  A  denominate  number  is  reduced  to  higher  denomi- 
nations by  division. 


134 


COMPOUND   NUMBERS 


/^ 

^^  ^^fe.         /--^ 

1'  ^ 

\ 

/^ 

^a^iN  Ml 

^^ 

x^  ^MM 

^^/-\ 

)p^ 

^  I'm 

^R 

r^^^^^^^F 

^«H 

T^ 

JKlALF  l||  ji^^ 

^^^^SsSSj^^BtB^mr  sN^HHI 

Hi'/   '^Hb 

.^&^i 

wOsHEd  i'^ 

QiPPsHiH  ii^hBIHh^K/'*?  ^HHB 

|f| 

10 

aHpj 

■H^ft' ' 

ii^Mii[H 

pusHEiyr 

liH  \\'^' 

*»       T^T^^^^ 

H^^ 

-^— "--", ." 

dticK^ 

HaMi*  iF  i 

J^^SB^BPS*^^^ 

:£ES==:. 

— -~-E^ 

n4i    Table  II*— Dry  Measure, 

The  denominations  bushels,  pecks,  quarts,  and  pints 
constitute  Dry  Measure,  They  are  used  in  measuring 
grain,  seeds,  fruits,  berries,  several  kinds  of  vegetables, 
lime,  charcoal,  and  some  other  articles. 

In  measuring  grain,  seeds,  peas,  beans,  and  small 
fruits,  the  measure  must  be  ex^en  full.  But  in  measuring 
large  fruits,  coarse  vegetables,  and  other  bulky  articles, 
the  measure  must  be  heaping  fall.  4  heaped  measures 
must  equal  5  even  measures. 

2  pt.  are  1  qt.  1  bu.  is  4  pk. 

8  qt.    "    1  pk.  (peck.)  1  pk.  "  8  qt.  . 

4  pk.  "    1  bu.  (bushel.)  1  qt.  "  2  pt. 

The  quart  and  pint  of  Dry  Measure  are  larger  than 
the  quart  and  pint  of  Liquid  Measure.  6  quarts  Dry 
Measure  are  equal  to  nearly  7  quarts  Liquid  Measure. 

EXEMCISJES. 

1.  Eead  17  bu.  1  pk.  2  qt.  1  pt. ;  2  pk.  1  pt. 

2.  Read  19  bu.  5  qt. ;  3  pk.  4  qt.  1  pt. 

3.  Write  one  bushel  two  pecks  four  quarts  one  pint. 

4.  Write  five  bushels  six  quarts  one  pint. 
6.  Write  twenty-eight  bushels  three  pecks. 


NOTATION    AND     REDUCTION.  135 

175.  Ex.  1.  Keduce  13  bu.  solution. 

3  qt.  to  quarts.  1|  ^^-  ^  P^'  ^  $^- 

Explanation. — Since  there  — 

are  no  pecks,  while  there  are  ^^  P'^' 

denominations  both  higher       

and  lower  than  pecks,  we  ^1^  +  3  =  419  qt. 

write   a  0  in  the  place   of  Hence,  13  Im.  3  qt.  =  419  qt. 
pecks  in  the  Solution. 

Ex.  2.  Eeduce  195  qt.  to  bu.  solution. 

Explanation.  —  In  reduc-         195  qt.  \  8  qt. 
ing  24    pecks    to    bushels,  24  pk.  Z  qt.  [4  pk, 

we  have  0  pecks  remaining.  ~^  , 

There  are  no  pecks,  there- 
fore, in  the  final  result.  ^ence,  195  qt.  =  6lu.3qt. 

176.   'Rules  for  deduction. 
I.   For  Reduction  Descending. 

1.  Multiply  the  highest  denomination  given  by  that  num- 
ber of  the  next  lower  denomination  which  equals  1  of  this 
higher,  and  to  the  product  add  the  given  lower  denomination. 

2.  In  the  same  manner,  reduce  this  result  to  the  next 
lower  denomination  ;  and  so  continue  until  the  given  num- 
ber is  reduced  to  the  required  denomination. 

n.  For  Reduction  Ascending. 

1.  Divide  the  given  denomination  by  that  number  of  this 
denomination  which  equals  1  of  the  next  higher,  writing 
the  quotient  as  so  many  of  the  higher  denomination,  and  the 
remainder  as  so  many  of  the  denomination  divided. 

2.  In  the  same  manner,  reduce  this  quotient  to  the  next 
higher  denomination  ;  and  so  continue  until  the  given  num- 
ber is  reduced  to  the  required  denomination. 

3.  Write  the  last  quotient  and  the  several  remainders  in 
their  order,  from  the  highest  denomination  to  the  lowed, 
for  the  required  result. 


136 


COMPOUND     NUMBERS 


jPH  OB  Jj  JEMS. 

18.  How  many  quarts  are  there  in  18  bu.  2  pk.  3  qt.  ?    595. 

19.  How  long  will  3  bu.  3  pk.  4  qt.  of  com  last  my  liens,  if 

1  feed  them  1  pt.  each  day  ?  18 4  days. 

20.  How  many  pint  papers  of  seed-corn  are  equal  to  7  bu. 
8  pk.  5  qt.  1  pt.  ? 

21.  Reduce  553  pints  to  higher  denominations. 

22.  A  farmer's  boy  fed  to  his  colt  1  pint,  of  oats  each  day 
for  150  days.     How  many  oats  did  he  feed  ?      21m.  1  ph.  3  qt. 

23.  A  blackberry  girl  sold  10  quarts  of  blackberries  each 
day  for  18  days.     How  many  berries  did  she  sell  ? 

5  lu.  2pk.  J^  qt. 

24.  Reduce  275  bu.  7  qt.  to  quarts.  8,807  quarts. 

25.  A  gardener  put  4  bu.  5  qt.  of  strawberries  into  quart 
boxes.     How  many  boxes  did  he  fill  ?  133. 

26.  How  many  times  can  a  pint  measure  be  filled  from  3  bu. 

2  pk.  1  pt.  of  chestnuts  ?  225. 

27.  Reduce  261  quarts  to  higher  denominations.     8  hu.  5  qt. 

28.  A  dealer  in  garden  seeds  put  up  353  pint  papers  of 
marrowfat  peas.    How  many  peas  did  he  put  up  ? 

51m.  2  ph  Ipt. 

29.  If  1  peck  of  clover  seed  will  seed  one  acre  of  land,  how 
much  land  can  a  farmer  seed  with  5.5  bushels  ?        22  acres. 

177t   Table  III,— Linear  Measure, 

The  denominations  miles,  rods,  yards,  feet,  and  inches 
constitute  Linear  or  Line  Measure.  They  are  used  in 
measuring  distances,  and  also  the  length,  width,  thick- 
ness, height,  and  depth  of  things. 

This  line  i.a.__^_i...ii_  is  one  inch  long. 


12     in.  (inches)  are  1  ft.  (foot.) 

3     ft.  "1  yd.  (yard.) 

5.5  yd.  "    1  rd.  (rod.) 

320     rd.  "1  mi.  (mile.) 


1  mi.  is  320     rd. 
1  rd.  "       5.5  yd. 
1  yd.  «      3     ft. 
1  ft.    "     12     in. 


NOTATION     AND     REDUCTION, 


137 


In  this  picture  tlie  gateway  is  represented  as  1  rod 
wide ;  the  board  fence  as  40  rods,  or  1  eighth  of  a  mile 
long ;  the  large  tree  as  80  rods,  or  1  fourth  of  a  mile 
from  the  comers ;  the  corners  as  160  rods,  or  1  half 
mile  from  the  gate ;  and  the  house  as  1  mile  from  the 
gate. 

EXEIt  CIS  ES. 

1.  Read  4  mi.  114  rd.  4  yd.  1  ft.  10  in. ;  17  mi.  46  rd.  3.5  ft. 

2.  Write  eight  miles  twenty-six  rods  two  yards  two  feet 
six  inches. 

3.  Write  three  hundred  nineteen  miles  sixty-seven  rods 
three  and  seventy-five  hundredths  yards. 


PR  OB  JL  JEMS. 


30.  How  many  feet 
are  63  rd.  4  yd.  2  ft.  ? 

31.  How  many  blocks 
each  1  inch  long  can  be 
cut  from  a  board  15  feet 
long?  180. 

32.  Reduce  5  mi.  187 
rd.  2  yd.  1  ft.  9  in.  to 
inches.     353,919  inches. 


SOLUTION   OF  PROBLEM 

63  rd.  4  yd.  %ft. 
5.5 

315 
315 


346.5  -t-  4  =  350.5  yd. 
3^ 

1051.5  +  2  =  1053.5 /i. 

Hence,  63  rd.  4  yd.  2  ft.  —  1053.5  ft. 


138 


COMPOUND     NUMBERS. 


33.  How  many  feet  are  200  mi.  4  yd.?  1,056,012. 

34.  How  many  rods  of  fence  will  it  take  to  inclose  a  farm 
winch,  is  1  mile  long  and  .5  of 


SOLUTION  OF    PROBLEM   36. 

398  yd.  1 5.5  yd. 
3980  [55 
385    ]J^rd. 

130 

110 

3.0  yd. 
Hence,  S98  yd.  =  72  rd.  2  yd. 


a  mile  wide  ?  960. 

35.  Reduce  398  yards  to 
rods. 

36.  In  a  bundle  of  lath  there 
are  100  pieces,  each  4  feet  long. 
K  laid  lengthwise  in  a  row 
upon  the  ground,  how  far 
would  they  reach  ? 

2JtTd.lyd.  1ft. 

37.  Reduce  1,530  inches  to  higher  denominations.    7  rd.  4  yd. 

38.  How  many  tiles  each  1  foot  long  will  be  required  for 
1  mi.  68  rd.  2  yd.  of  tile-drain  ?  6,408. 

39.  How  many  miles  are  there  in  the  fences  that  inclose  the 
farm  shown  in  the  map  on  page  25  ?  2  mi.  242  rd. 

178«   Table  IV.— Square  Measure. 

The  denominations  square  miles,  acres,  square  rods, 
square  yards,  square  feet, 
and  square  inches  consti- 
tute Square  Measure.  They 
are  used  in  measuring 
land,  flooring,  plastering, 
and  other  surfaces. 

A  square  foot  is  1  foot 
or  12  inches  long,  and  1 
foot  or  12  inches  wide. 
Hence,  it  contains  12  times 
12,  or  144  square  inches. 

144       sq.  in.  (square  in.)  are  1  sq.  ft. 

9   sq.  ft.        "  1  sq.  yd. 
30.25  sq.  yd.       "  1  sq.  rd. 
160   sq.  rd.       "  1  A.  (acre.) 
640   A.  "1  sq.  mi. 


w.. 

g 

& 

zn 

^ 
^ 

(itt 

12  inebLes  wide 

1  sq.  mi.  is  640   A. 
1  A.    "  160   sq.  rd. 
Isq.rd.  "  30.25  sq.  yd. 
1  sq.  yd.  "   9   sq.  ft. 
1  sq.  ft.  "  144   sq.  in. 


NOTATION    AND    BEDUCTION. 


139 


EXEMCISES. 

1.  Bead  14  sq.  mi.  84  A.  28  sq.  rd. 

2.  Read  25  sq.  rd.  16  sq.  yd.  84  sq.  in. 

3.  Write  two  hundred  nine  square  miles  eighty  six  acres 
one  hundred  seven  square  rods. 

4.  Write  five  square  yards  eight  square  feet  thirty-six 
rquare  inches. 

PMOBJOEMS. 

40.  Reduce  34  square  miles  to  square  rods.  3,431,600  sq.  rd. 

41.  A  farmer  planted  one  hill  of  com  upon  every  square 
yard  of  a  10-acre  lot.     How  many  hills  did  he  plant  ?  48,400. 

42.  Reduce  84  sq.  rd.  4  sq.  ft.  to  square  feet.     22,873  sq.ft. 

43.  In  25.3  square  miles  how  many  acres  ?  16,192  A. 

44.  How  many  square  miles  are  there  in  312,000  square  rods  ? 

45.  A  fruit  grower  has  an  orchard  containing  6,386  peach 
trees,  and  each  tree  occupies  1  square  rod  of  land.  How 
much  land  is  there  in  the  orchard  ?  39  A.  I46  sq.  rd. 

46.  In  covering  a  roof  a  tinsmith  used  1,152  sheets  of  tin, 
each  covering  14  by  20  inches.  How  many  square  feet  were 
in  the  roof?  2,240. 

47.  Reduce  334,976  square  inches  to  higher  denomina- 
tions. .  8  sq.  rd.  16  sq.  yd.  4  sq.ft.  32  sq.  in. 

179.   Table  V,— Cubic  Measure. 

The  denominations  cu- 
bic yards,  cubic  feet,  and 
cubic  inches  constitute 
Cubic  Measure.  They  are 
used  in  measuring  earth, 
timber,  stone,  and  many 
other  articles,  and  in  esti- 
mating the  capacity  of 
bins,  boxes,  etc. 

A  cubic  foot  is  1  foot 
or  12  inches  long,  1  foot 
or  12  inches  wide,  and 
1     foot     or     12     inches 


140  COMPOUND     NUMBERS. 

thick,  and  lience  it  contains  12  times  12  times  12,  or 
1728,  cubic  inches.  A  cubic  yard  is  3  feet  long,  3  feet 
wide,  and  3  feet  tliick,  and  contains  27  cubic  feet. 

1738  cu.  in.  (cubic  incli)  are  1  cu.  ft.  l  1  cii.  yd.  is      27  cu.  ft. 
27  cu.  ft.  "   1  cu.  yd.  |  1  cu.  ft.    "  1728  cu.  iu. 

EXEB,  CISE8. 

1.  Read  30  cu.  yd.  10  cu.  ft.  1008  cu.  in. 

2.  Read  215  cu.  yd.  49  cu.  in. 

3.  Write  fourteen  cubic  yards  twenty-four  cubic  feet. 

4.  Write  one  hundred  nine  cubic  yards  ninety-two  cubic 
inches. 

PItOnLE3IS. 

48.  In  3  cu.  yd.  17  cu.  ft.  112  cu.  in.,  how  many  cubic  inches  ? 

49.  Reduce  846,296  cubic  inches  to  higher  denominations. 

18  cu.  yd.  3  cu.  ft.  130 Jf,  cu.  in. 

50.  Reduce  5  cu.  yd.  948  cu.  in.  to  cubic  inches.      23Jf,228. 

51.  How  many  cubical  blocks,  each  containing  1  cubic 
inch,  will  be  required  to  make  a  pile  that  shall  contain  16.5 
cubic  yards?  769,824. 

52.  A  brick  is  8  inches  long,  4  inches  wide,  and  2  inches 
thick.  If  100,000  bricks  are  piled  together,  how  many  cubic 
yards  will  there  be  in  the  pile  ?   137  cu.  yd.  4  cu.  ft.  1216  cu.  in. 

53.  Reduce  4,713,256  cubic  inches  to  higher  denominations. 

101  cu.  yd.  1000  cu.  in. 

180.   Table  VI,— Wood  3Ieasure, 

The  denominations  cords,  cord  feet,  and  cubic  feet 
constitute  Wood  Measure.  They  are  chiefly  used  in 
measuring  wood.  Kough  stone  is  also  commonly 
sold  by  the  cord. 

A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet 
high  contains  1  cord ;  and  1  foot  in  length  of  such  a 
pile  contains  16  cubic  feet,  and,  therefore,  is  1  cord 
foot. 


NOTATION     AND     REDUCTION. 


141 


16  cu.  ft.  are  1  cd,  ft.  (cord  foot.) 

8  cd.  ft.  "    1  cd.  (cord.) 
128  cu.  ft.  "    1  cd. 


1  cd.  is  8  cd.  ft. 
1  cd.  '^  138  cu.  ft. 
1  cd.  ft.  "     16  cu.  ft. 


:EXEItCISES. 

1.  Eead  7  cd.  5  cd.  ft.  8  cu.  ft. ;  19  cd.  4  cd.  ft. 

2.  Write  twenty  cords  seven  cord  feet  six  cubic  feet. 

3.  Write  two  hundred  fifty-one  cords  two  cord  feet. 

PMOBLEMS. 

54.  How  many  cubic  feet  are  13  cd.  5  cd.  ft.  12  cu.  ft.  ?  1,756. 

55.  In  2,240  cubic  feet  of  cobble  stone,  how  many  cords  ? 


142 


COMPOUND     NUMBERS. 


56.  How  many  cords  of  wood  in  a  pile  76  feet  long,  4  feet 
wide,  and  4  feet  high  ?  9.5. 

57.  How  much  wood  is  there  in  a  pile  120  feet  long,  4  feet 
wide,  and  6  feet  high  ?  22.5  cords. 

58.  How  much  wood  will  a  teamster  draw  at  10  loads,  if 
each  load  is  12  feet  long,  4  feet  wide,  and  3  feet  high  ? 

59.  If  a  pile  of  wood  is  4  feet  wide  and  4  feet  high,  how 
long  must  it  be  to  contain  56.25  cords  ?  450  feet. 


181.    Table    VII.— Weight, 

Most  kinds  of  produce,  provisions,  and  groceries, 
also  iron  and  other  metals,  coal  and  many  other  arti- 
cles, are  bought  and  sold  by  Weight.  The  denomina- 
tions in  common  use  are  tons,  hundred-weight,  pounds, 
and  ounces. 

16  oz.  (ounces)  are  1  lb.  (pound.)  1  T.    is   20  cwt. 

100  lb.  "    1  cwt.  (hundred-weight.)  1  c^i;.  "  100  lbs. 

20  cwt.  "    1  T.  (ton.)  1  lb.    "    16  oz. 

200  lb.  of  pork,  beef,  or  fish  are  1  bbl.  (barrel.) 

196  1b.  of  flour  are  1  bbl. 


NOTATION     AND     REDUCTION.  143 

EXER  CISES. 

1.  Read  5  T.  17  cwt.  34  lb.  4  oz. ;  39  T.  94  lb. 

2.  Write  seven   tons    thirteen    hundred-weight    fifty-nine 
pounds  fourteen  ounces. 

3.  Write  one  ton  nine  hundred  forty-eight  pounds  five 
ounces. 

PMOBIjEMS. 

60.  How  many  ounces  are  there  in  9  tons  ?  288,000. 

61.  Reduce  4  T.  16  cwt.  83  lb.  to  pounds.  9,683. 
63.  One  month  a  manufacturer  put  up  5  T.  4  cwt.  39  lb.  of 

saleratus  in  pound  packages.    How  many  packages  did  he 
put  up  ?  10,429. 

63.  How  many  pound  bars  of  lead  will  weigh  3  T.  54  lb.  ? 

64.  How  much  will  6  barrels  of  mackerel  cost  at  $.085  a 
pound  ?  $102. 

65.  A  grocer  retailed  13  barrels  of  flour  at  $.055  a  pound. 
How  much  did  he  receive  for  it  ?  $140.14. 

66.  How  many  tons  are  34,000  pounds  ?  17  T. 

67.  If  1  oz.  of  lead  is  used  in  making  1  rifle  ball,  how  much 
lead  will  be  required  to  make  60,000  balls  ?    1  T.  17  cwt.  50  lb. 

68.  How  many  barrels  of  flour  are  6,860  pounds  ?         35. 

69.  Reduce  64,015  oz.  to  higher  denominations.    2  T.  15  oz. 

182.   Table  VIII.— Counting. 

In  counting  articles  for  market  purposes,  the  de- 
nominations dozen  and  gross  are  used. 

13  ones  are  1  doz.  (dozen.)    I  1  gro.  is  13  doz. 

13  doz.    "    1  gro.  (gross.)     I  1  doz.  "  13  ones. 

EXEItCISES. 

1.  Read  24  gro.  5  doz. ;  29  gro.  7  doz. 

2.  Write  seventeen  gross  six  dozen. 

3.  Write  fifty  gross  nine  dozen. 


144  .COMPOUNDNUMBERS. 

FJROBJLEMS. 

70.  If  a  cook  uses  6  eggs  each  day,  in  how  many  days  will 
she  use  9  doz.  eggs  ?  18. 

71.  How  many  steel  pens  are  there  in  7  boxes  each  con- 
taining 1  gro.  ?  1,008. 

72.  How  many  clothes-pins  in  47  doz.  ?  561^. 

73.  How  many  dozens  are  132  brooms  ?  11. 

74.  In  one  year  a  tailor  used  23  gro.  9  doz.  buttons.  What 
number  of  buttons  did  he  use  ?  3,420. 

75.  If  18  crayons  are  used  every  week  in  a  certain  school, 
how  many  gross  will  be  used  in  40  weeks  ?  5. 

76.~  In  1,844  screws,  how  many  gross  ?         12  gro.  9  doz.  8. 

183.    Table  IX.— Paper. 

Paper  is  bought  and  sold  by  the  ream,  quire,  and 
sheet. 

24  sheets  are  1  quire.  I  1  rm.      is  20  quires. 

20  quires  "    1  rm.  (ream.)    I  1  quire  "  24  sheets. 

EXJEIt  CIS  ES. 

1.  Bead  4  rm.  8  quires  12  sheets  ;  11  rm.  10  quires. 

2.  Write  thirteen  reams  fifteen  quires  four  sheets. 

PR  OBLEMS. 

77.  How  many  sheets  of  paper  are  there  in  a  ream  of  fools- 
cap ?  480- 

78.  If  1  sheet  of  printing  paper  will  make  4  handbills,  how 
many  bills  will  5  rm.  8  quires  make  ?  10,368. 

79.  How  many  letters,  each  requiring  1  sheet,  can  be  written 
on  11  rm.  12  quires  of  commercial  note  paper  ?  6.,568. 

80.  In  384  sheets  of  letter  paper,  how  many  quires  ?      16. 

81.  If  a  lawyer  uses  18  sheets  of  legal  cap  paper  every  day, 
how  many  reams  will  he  use  in  320  days  ?  12. 

82.  If  a  merchant  uses  3  quires  of  wrapping  paper  every 
day  of  the  313  week-days  of  the  year,  how  much  paper  does 
he  use  in  the  year  ?  46  rm.  19  quires. 


NOTATION     AND     REDUCTION. 


145 


184.   Table  X,—Time, 

The  denominations  centuries,  years,  months,  weeks, 
days,  hours,  minutes,  and  seconds  are  used  in  express- 
ing different  portions  of  time.  The  day  and  the  year 
are  the  natural  divisions  of  time,  the  other  denomina- 
tions, except  centuries,  being  parts  of  these. 

60  sec.  (seconds)  are  1  min.  (minute.) 


60  min. 
24  h. 

7  da. 

52  wk.  1  da. 

or  365  da. 

52  wk.  2  da. 

or  366  da. 

100  yr. 


are  1  h.     (hour.) 
"  1  da.  (day.) 
"  1  wk.  (week.) 
1  common 
yr.  (year.) 


(" 


1  leap-yr. 
1  century. 


1  century 
1  leap-year 

1  common  yr.  " 

1  da.  " 

Ih.  " 

1  min.  " 


is  100  yr. 
u  \  52  wk.  2  da. 
I     or  366  da. 
52  wk.  1  da. 
or  365  da. 
24  h. 
60  min. 
60  sec. 


Every  fourth  year  from  the  beginning  of  a  century  is 
a  leap-year. 

EXEIt  CIS  ES. 

1.  Read  3  yr.  6  mo. ;  12  h.  30  min.  15  sec. 

3.  Read  9  wk.  3  da.  10  h. ;  4  da.  15  h.  45  min. 

3.  Write  five  hours  fifteen  minutes  thirty  seconds. 

4.  Write  fourteen  weeks  six  days  four  hours. 
6.  Write  twenty-eight  years  nine  months. 

M 


146 


COMPOUND     NUMBERS. 

Seasons. 

Months. 

Bays. 

Winter. 

(    1st  a 
I    2d     ' 

10.        January, 

Jan. 

31 

'          February, 

Feb. 

28 

(    3d      ' 

'          March, 

Mar. 

31 

Spring. 

■|    4th    ' 

'          April, 

Apr. 

30 

(    5th    ' 

May, 

May. 

31 

C    6th    ' 

'          June, 

June. 

30 

Summer. 

]    7th    ' 

July, 

July. 

31 

i    8th    * 

'          August, 

Aug. 

31 

(    9th    ' 

'          September, 

Sept. 

30 

Autiimn. 

KlOth    ' 

'          October, 

Oct. 

31 

Cllth    ' 

'          November, 

Nov. 

30 

Winter. 

12th    ' 

'          December, 

Dec. 

31 

February  iias  28  days  in  a  common  year,  and  29  in  a 

leap-year.  (See  Manual,  page  219.) 

83.  How  many  minutes  in  the  month  of  January  ?      4.4,640. 

84.  How  many  seconds  are  there  in  a  common  year  ? 

81,536,000. 

85.  Reduce  3  wk.  20  min,  to  minutes.  80,260  min. 

86.  Reduce  50,400  minutes  to  weeks.  5  weeks. 

87.  How  many  seconds  in  the  three  summer  months  ? 

7,948,800. 

88.  How  long  will  it  take  a  clock  to  tick  1,000,000  times, 
if  it  ticks  once  every  second  ?      1  wk.  4  da.  13  h.  46  min.  40  sec. 

89.  How  many  days  were  there  from  the  beginning  of  the 
year  1857  to  the  end  of  the  year  1866,  two  of  the  years,  1860 
and  1864,  bemg  leap-years  ?  3, 652. 

90.  In  a  leap-year,  how  many  hours  ?  8,784. 

91.  Reduce  875,665  sec.  to  higher  denominations. 

10  da.  3  Ti.  14  min.  25  sec. 

92.  After  the  9th  day  of  October,  how  many  hours  remain 
in  the  month  ?  528. 

93.  If  you  can  count  75  every  minute,  how  much  time  would 
you  spend  in  counting  27,000,000  ?  35  wk.  5  da. 


NOTATION     AND     REDUCTION.  147 


185.  The  Metric  System  of  Weights  and  Measures. 

In  the  year  1866,  the  Congress  of  the  United  States 
passed  a  bill  authorizing  the  use  of  a  new  system  of 
weights  and  measures.  In  this  system  the  principal 
denomination  is  the  Metre,  from  which  all  the  other 
denominations  in  all  the  tables  are .  derived.  Hence, 
this  system  is  called  the  Metric  System. 

The  principal  denomination  for  the  Measure  of  Sur- 
face is  the  Are;  for  the  Measure  of  Capacity,  the  Litre ; 
and  for  Weight,  the  Gram.    (See  Manual,  page  219.) 

The  lower  denominations  in  each  table  are  tenths, 
hundredths,  or  thousandths  of  these  ;  and  their  names 
are  formed  by  prefixing  deci,  centi,  or  mUli  to  the  name 
of  the  principal  denomination. 

The  higher  denominations  are  10,  100,  1,000,  or 
10,000  times  the  principal  denomination  of  any  table ; 
and  their  names  are  formed  by  prefixing  deJca,  hecto, 
lilOy  or  myria  to  the  name  of  that  principal  denomina- 
tion. 

TABLE   OF   DENOMINATIONS   AND   THEIB   EELATIVE   VALUES. 


PREFIXES  FOR 
LOWEB  DENOMINATIONS. 

MilU  (mill-ee)  .001  of 
Centi  (sent-ee)  .01  of 
Ded  (des-ee)   .1      of 


NAMES   OP 

PKINCIPAL 

DENOMINATIONS. 

Metre  (mee-ter) 
Are       (are) 
Litre   (li-ter) 
Gram 


PREFIXES  FOR 
HIGHER  DENOMINATIONS. 

BeJca   (dek-a)  10 

Eecto  (hec-to)  100 
Kilo  (kill-o)  1,000 
Myria  (mir-e-a)  10,000 


The  weights  and  measures  of  this  system  have  not 
yet  come  into  use  in  this  country.  They  are  in  general 
use  in  France,  Belgium,  Spain,  and  Portugal ;  and 
their  use  has  been  legaHzed  by  Great  Britain,  Italy, 
Norway,  Sweden,  Greece,  Mexico,  and  most  of  the 
South  American  governments. 


148 


COMPOUND  NUMBERS, 


UEES  OF 


10  millimetres  are 
10  centimetres  " 
10  decimetres  " 
10  metres  " 

10  dekametres  " 
10  hectometres" 
10  kilometres  " 


1  centimetre 
1  decimetre 
1  metre 
1  dekametre 
1  hectometre 
1  kilometre 
1  myriametre 


LENGTH. 

millimetre 
centimetre 
decimetre 

METRE 

dekametre 
hectometre 
kilometre 
myriametre 


y-oV(T  metre 
jh    metre 
yV      metre 
S9.37  inches. 
10         metres 
100       metres 
1,000    metres 
10,000  metres 


MEASURES   OF   SURFACE. 


100  centares  are  1  are 
100  ares  "   1  hectare 


1  centai'e 

1  ARE 

1  hectare 


is  y^o  are 

{( j  100  sq.  metres, 

( 119.6  sq.  yd. 
"  100  ares 


MEASURES   OF   CAPACITY. 


10  millilitres  are 
10  centilitres    " 
10  decilitres     '' 

10  litres 


1  centilitre 
1  decilitre 
1  litre 

1  dekalitre 


10  dekalitres 
10  hectolitres 


1  hectolitre 


r 


millilitre  is  y^jVo  ^^^^^ 
litre 
litre 


1  centilitre  "  jli 
1  decilitre   "  j\ 

fl  cu.  decimetre 

1  LITRE         "I    .908  qt.  dry  meas. 
[  1.0567  qt.liq.meas. 

1  dekalitre  "  10      litres 


kilolitre,  or  1  hectolitre  "  100    litres 


\^^IGHT. 


10  milligrams  are  1  centigram 
decigram 
gram 
dekagram 
hectoOTam 


10  centigrams  " 
10  decigrams  " 
10  grams  " 

10  dekagrams  " 
10  hectograms  " 
10  kilograms  j.  ^^ 

or  kilos     ) 
10  myriagrams" 

10  quintals       " 


kilogram 

myriagram 

quintal 
millier 
or  tonneau 


1  milligram 
1  centigram 
1  decigram 

1  GRAM 

1  dekagram 
1  hectogram 
1  kilogram) 
or  1  kilo) 
1  myriagram 
1  quintal 
1  millier 


f oVo  gram 
tIo    gram 
TO      gram 
15.432  grains 
10      grams 
100    grams 
1000  grams,  or 
2.2046  pounds 
10      kilos 
100    kilo^ 
1,000  kilos 


ADDITION.  149 

SECTION  II. 

186.  Ex.  What  is  the  sum  of  5  rd.  4  yd.  2  ft.  3  in., 
7  rd.  1  yd.  1  ft.  9  in.,  2  yd.  1  ft.,  2  ft.  11  in.,  and  12  rd. 
5  yd.  5  in.  ? 

Explanation. — We  write  the                  solution. 
numbers  so  that  like  denomina-       5  ^^^  4  r^.^  2  ft.  3  in. 
tions  stand  in  the  same  columns.       7        119 
Commencing   with    the   lowest                2         10 
denomination,   we   add ;   5  in.                         2     11 
-t-  11  in.  +  0  in.  +  9  in.  +  3  in.     1? 5_ 9__? 


=:  28  in.  But  28  in.  =:  2  ft.  4  26  rd.  3  yd.  2  ft.  4  in. 
in. ;  we  therefore  write  the  4  in. 

as  the  inches  of  the  sum,  and  the  2  ft.  we  add  with  the 
feet  of  the  given  numbers.  2  ft.  +  0  ft.  +  2  ft.  +  1  ft. 
+  1  ft.  +  2  ft.  =  8  ft.  But8ft.  =r  2  yd.2ft  ;  we  there- 
fore write  the  2  ft.  in  the  sum,  and  add  the  2  yd.  with 
the  column  of  yards.  2  yd.  +  5  yd.  +  2  yd.  +  1  yd.  + 
4  yd.  =  14  yd.  But  since  14  yd.  =  2  rd.  3  yd.,  we  write 
the  3  yd.  in  the  sum,  and  add  the  2  rd.  with  the  column 
of  rods.  2  rd.  4-  12  rd.  +  7  rd.  +  5  rd.  =  26  rd., 
which  we  write  as  the  rods  of  the  sum.  The  result, 
28  rd.  3  yd.  2  ft.  4  in.,  is  the  sum  required. 

PBOBZEMS. 

Find  the  sum  of  the  compound  numbers  in  problems  1,  2,  3. 

(1)  (3)  (3) 

9  A.  96  sq.  rd.  3  rm.  5  quires  16  sheets  34  gal.  2  qt.  0  jJt. 

11       44  4         0  8  35         1        1 

8     108  3       20  6  33         3       1 

10       56  2       18  14  36         1       0 


4.  A  painter  used  5  gal.  3  qt.  1  pt.  of  linseed-oil  one  week, 
and  3  gal.  2  qt.  1  pt.  the  next  week.  How  much  did  he  use 
in  the  two  weeks  ?  9  gal.  2  qt. 


150  COMPOUND     NUMBERS. 

5.  A  farmer  used  3  bu.  1  pk.  6  qt.  of  clover  seed  in  seed- 
ing one  field,  and  3  bu.  3  pk.  4  qt.  in  seeding  another.  How 
much  did  he  use  upon  the  two  fields  ?  6  bu.  1  ph  2  qt. 

6.  How  much  wood  is  there  in  three  piles,  the  first  of  which 
contains  5  cd.  3  cd.  ft.  13  cu.  ft.,  the  second  6  cd.  6  cd.  ft.,  and 
the  third  9  cd.  4  cd.  ft.  4  cu.  ft.  ?  21cd.6  cd.ft. 

7.  A  father  is  28  yr.  164  da.  older  than  his  son,  and  the  son 
is  17  yr.  235  da.  old.     How  old  is  the  father  ?    46  yr.  24  da. 

187.    !%igU  for  A.ddltion  of  Compound  JVumbers, 

I.  Write  the  parts  with  like  denominations  in  the  same 
column. 

n.  Add  each  denomination  separately,  beginning  with 
the  lowest ;  and  when  the  sum  is  less  than  1  of  the  next 
higher  denominatioUj  write  it  under  the  denomination 
added. 

m.  When  the  sum  of  any  denomination  is  equal  to  or 
more  than  1  of  the  next  higher  denomination,  reduce  it  to 
that  higher  denomination,  write  the  remainder  under  the 
denomination  added,  and  add  the  quotient  with  the  next 
higher  denomination. 

PM  OBIjEMS. 

8.  A  stationer  sold  5  gro.  3  doz.  8  pens  of  one  kind,  and  2 
gro.  6  doz.  6  pens  of  another.     How  many  pens  did  he  sell  ? 

7  gro.  10  doz.  2. 

9.  A  livery-man  bought  3  loads  of  hay,  the  first  weighing 
1  T.  2  cwt.  17  lb.,  the  second  1  T.  3  cwt.  96  lb.,  and  the  third 
19  cwt.  49  lb.     How  much  hay  did  he  buy  ?     ST.  5  cwt.  62  lb. 

10.  A  housekeeper  made  2  gal.  1  qt.  1  pt.  of  currant  wine, 
1  gal.  3  qt.  of  blackberry  wine,  and  4  gal.  2  qt.  1  pt.  of  grape 
wine.     How  much  wine  did  she  make  ?  8  gal.  3  qt. 

11.  In  five  successive  days,  a  fruit  dealer  sold  3  pk.  3  qt.  1 
pt.  of  cherries,  1  bu.  1  pk.  5  qt.,  1  bu.  3  pk.  7  qt.,  1  bu.  2  qt. 
1  pt.,  and  3  pk.  1  pt.     How  many  cherries  did  he  sell  ? 

12.  What  is  the  sum  of  5  rm.  14  quires  12  sheets,  7  rm.  11 
quires  9  sheets,  and  9  rm.  15  quires  9  sheets  ? 


ADDITION.  151 

13.  A  railroad  train  runs  from  Detroit  to  Ann  Arbor  in  1  h. 
45  min. ;  to  Jackson  in  1  b.  40  min.  more ;  to  Marshall  in  1 
h.  25  min.  more ;  to  Kalamazoo  in  1  h.  35  min.  more ;  to 
Mies  in  3  h.  15  min.  more ;  and  to  Chicago  in  4  h.  30  min. 
more.  What  is  the  running  time  of  the  train  from  Detroit  to 
Chicago  ?  13  h.  10  min. 

14.  In  an  ax  factory  are  six  large  grindstones,  which  weigh 
2  T.  1  cwt.  18  lb.,  1  T.  16  cwt.  24  lb.,  2  T.  3  cwt.  7  lb.,  2  T. 
2  cwt.  7  lb.,  1  T.  18  cwt.  87  lb.,  and  1  T,  19  cwt.  69  lb.  What 
is  their  total  weight  ?  12  T.  1  cwt.  12  lb. 

15.  A  telegraph  company  put  up  1  mi.  14  rd.  3  yd.  of  wire 
one  day,  318  rd.  5  yd.  the  second  day,  1  mi.  39  rd.  4  yd.  the 
third  day,  and  1  mi.  67  rd.  the  fourth  day.  How  much  wire 
did  they  put  up  in  the  four  days  ?  4  mi.  120  rd.  1  yd. 

16.  What  is  the  sum  of  9  cu.  yd.  20  cu.  ft.  388  cu.  in.,  218 
cu.  yd.  14  cu.  ft.  524  cu.  in  ,  and  145  cu.  yd.,  11  cu.  ft.  1415 
cu.  in.  ?  373  cu.  yd.  19  cu.ft.  599  cu.  in. 

17.  There  are  35  sq.  yd.  5  sq.  ft.  of  plastering  in  the  ceiling 
of  a  room,  22  sq.  yd.  2  sq.  ft.  in  each  of  the  two  side  walls,  and 
17  sq.  yd  7  sq.  ft.  in  each  of  the  two  end  walls.  How  much 
plastering  in  the  room  ?  115  sq.  yd.  5  sq.ft. 

18.  Add  4  yd.  2  ft.  4  in.,  3  yd.  1  ft.  8  in.,  5  yd.  2  ft.  6  in. 

19.  Add  28  wk.  4  da.  14  h.  45  min.  45  sec,  11  wk.  3  da. 
10  h.  30  min.  15  sec,  and  6  wk.  6  da.  3  h.  25  min.  30  sec. 


SECTION   III. 

S  VS  T^A.  C  TIOJV. 


188.  Ex.  1.  From  8  bu.  1  pk.  7  qt.  subtract  4  bu. 
3  pk.  2  qt. 

Explanation. — We  write  the  de- 
nominations of  the  subtrahend  un-  solution. 
der  the  Hke  denominations  of  the      ?^^-  \^^'  I 
minuend.     Commencing  with    the 


4         3         2 


lowest  denomination,  we  subtract;      ^^^-  ^P^-  ^9^- 


152  COMPOUND     XUMBEKS. 

2  qt.  from  7  qt.  leave  5  qt.,  wliicli  we  write  as  the 
quarts  of  the  remainder.     Since  we  can  not  subtract 

3  pk.  from  1  pk.,  we  take  1  bu.  (=4  pk.)  of  the  8  bu. 
in  the  minuend,  add  it  to  the  1  pk.,  and  from  the  5  pk. 
thus  obtained,  we  subtract  the  3  i)k.,  writing  the  dif- 
ference, 2  pk.,  in  the  remainder.     Finally,  we  subtract 

4  bu.  from  the  7  bu.  now  left  in  the  minuend,  and  write 
the  difference,  3  bu.,  in  the  remainder.  The  compound 
number,  3  bu.  2  pk.  5  qt.,  is  the  remainder  required. 

Ex.  2.  From  20  gal.  subtract  5  gal.  2  qt.  1  pt. 
Explanation. — Since  we  can  not 
subtract  1  pt.  from  0  pt.,  and  there  solution. 

are  no  quarts  in  the  minuend  to      i^ ? ? 

reduce  to  pints,  we  take  1  of  the       ^^  ^al'  ^   t    1    t 

20  gal.,  leaving  19  gal.     From  this       — — l^^JPjL. 

1  gal.  (or  4  qt.)  we  take  1  qt.,  leav-       14  (/aZ.  1  qt.  Ipt. 
3  qt.  ;  and  this  1  qt.  ==  2  pt.     The 
form  of  the  minuend  is  now  changed  from  20  gal.  to 
19  gal.  3  qt.  2  pt.,  and  from  this  we  subtract  5  gal.  2  qt. 
1  pt.,  obtaining  a  remainder  of  14  gal.  1  qt.  1  pt. 

/^N         PJtOBZEMS.  /2) 

From  7  da.  3  h.  20  min.  5  mi.  220  rd.  4  yd.  2ft.  5  in. 
Subtract  3         9       15  2        264       3 2_^_ 

3.  From  a  cask  that  contained  33  gal.  2  qt.  of  vinegar,  a 
erocer  drew  17  gal.  3  qt.  How  much  vinegar  was  left  in  the 
Lk?  15  gal  3  qt. 

4.  A  farmer  raised  614  bu.  1  pk.  of  oats,  and  sold  all  but 
133  bu.  3  pk.     How  many  oats  did  he  sell  ?       4^0  hi.  2  ph 

5.  A  physician  bought  a  load  of  hay  which  weighed,  with 
the  wagon,  1  T.  8  cv/t.  21  lb.  The  wagon  alone  weighed  12 
cwt.  43  lb.     What  was  the  weight  of  the  hay  ?      15  cwt.  78  lb. 

6.  A  merchant  tailor  bought  32  gro.  6  doz.  rubber  buttons, 
and  sold  24  gro.  8  doz.  6.     How  many  buttons  had  he  left  ? 


SUBTRACTION.  153 

7.  The  walls  of  a  room  measure  68  sq.  yd.  4  sq.  ft.,  and  the 
windows  and  doors  18  sq.  yd.  7  sq.  ft.  How  many  square 
yards  of  plastering  on  the  walls  ?  ^9  sq.  yd.  6  sq.ft. 

8.  A  farmer  exchanged  a  farm  of  200  acres  for  another  con- 
taining 113  A.  38  sq.  rd.  How  much  more  land  was  there  in 
the  first  farm  than  in  the  second  ?  86  A.  132  sq.  rd. 

9.  A  bookseller  having  23  rm.  13  quires  of  letter-paper, 
sold  13  rm.  16  quires  13  sheets.  How  much  paper  had  he 
then  ?  8  rm.  15  quires  12  sheets. 

10.  A  grocer  bought  a  crock  of  butter  which  weighed  44  lb. 
6  oz.  The  crock  alone  weighed  7  lb.  10  oz.  How  much  did 
the  butter  weigh  ?  86  lb.  12  oz. 

11.  A  druggist  put  5  gal.  3  qt.  1  pt.  of  alcohol  into  a  can 
which  would  hold  30  gal.  How  much  more  alcohol  would 
the  can  have  held  ?  IJf,  gal.  1  qt.  1  pt. 

12.  A  laborer  agreed  to  dig  a  cellar  18  ft.  long,  16  ft.  wide, 
and  5  ft.  deep.  After  digging  44  cu.  yd.  of  earth,  how  much 
more  has  he  to  remove  ?  9  cu.  yd.  9  cu.ft. 

13.  William  is  16  yr.  28  da.  old,  and  Edward  is  11  yr.  284  da. 
old.    How  much  older  is  William  than  Edward  ?    4  yr.  109  da. 

14.  A  farmer  contracts  to  deliver  at  a  railroad  station  1,000 
cords  of  wood.  He  has  384  cd.  5  cd.  ft.  already  cut.  How 
much  more  wood  must  he  chop  ?  615  cd.  S  cd.  ft. 

15.  From  90  cu.  yd.  subtract  39  cu.  yd.  18  cu.  ft.  966  cu.  in. 

50  cu.  yd.  8  cu.  ft.  762  cu.  in. 
Ex.   3.    How   many  years,  months,  and  days  from 
April  15,  1864,  to  July  4,  1867  ? 

Explanation.  —  Since    the  solution. 

later  of  two  dates  is  expressed      ISQT  yr.  7  mo.     4:  da. 

by  a  greater  compound  num- 

ber  than  the  earlier,  we  write  3  yr.  2  mo.  19  da. 

the  later  date  for  the  minuend, 

and  the  earlier  for  the  subtrahend,  writing  the  number 
of  the  year,  month,  and  day  in  order.  We  then  sub- 
tract as  in  other  compound  numbers,  calling  30  days  a 
month  when  the  number  of  days  in  the  subtrahend  is 
greater  than  that  in  the  minuend. 


164  COMPOUNDNUMBERS. 

189.   "Rule  for  Subtraction  of  Compound  JVumbers, 

.  I.   Write  the  denominations  of  the  subtrahend  under  the 
like  denominations  of  the  minuend. 

II.  Subtract  each  denomination  of  the  subtrahend  from 
the  like  denomination  of  the  minuend,  and  write  the  result 
as  the  same  denomination  in  the  remainder. 

III.  WJien  any  denomination  of  the  subtrahend  exceeds 
that  in  the  minuend,  before  subtracting,  reduce  1  of  the  next 
higher  denomination  of  the  minuend  to  this  lower  denomi- 
nation, and  add  it  to  the  number  of  this  denomination  given 
in  the  minuend. 

rV.  In  the  last  case,  in  subtracting  the  next  higher  de- 
nomination, we  may  either  call  the  number  in  the  minuend 
1  less,  or  that  in  the  subtrahend  1  more. 

jpm  oblems. 

16.  Benjamin  Franklin  was  born  Jan.  17,  1706,  and  died 
Apr.  17,  1790.     How  old  was  he  when  lie  died  ?     SJj.  yr.  3  mo. 

17.  George  "Washington  was  born  Feb.  32,  1732,  and  died 
Dec.  14,  1799.     At  what  age  did  he  die  ?     67  yr.  9  mo.  22  da. 

18.  A  note  dated  June  7, 1863,  was  paid  Apr.  4,  1865.  How 
long  did  it  remain  unpaid  ?  1  yr.  9  mo.  27  da. 

19.  A  note  was  given  Sept.  10,  1867,  payable  Feb.  4,  1868. 
How  long  had  it  to  run  ?  J^  mo.  2^  da. 

20.  Robert  was  bom  Oct.  9,  1858.  How  old  was  he,  May 
11,  1867  ?  8yr.7  mx).2  da. 

21.  Washington  Irving  died  Nov.  28, 1859,  aged  76  yr.  7  mo. 
25  4a.     What  was  the  date  of  his  birth  ?  J._p7*.  3,  1783. 

22.  A  farmer  in  a  country  district  agrees  to  furnish  10  cd. 
4  cd.  ft.  of  wood  for  the  winter  term  of  school.  After  drawing 
4  cd.  7  cd.  ft.,  how  much  has  he  yet  to  draw  ?      5  cd.  5  cd.ft. 

23.  From  11  mi.  84  rd.  4  yd.  1  ft.  take  5  mi.  186  rd.  2  yd. 
3  ft.  5  mi.  218  rd.  1  yd.  2  ft. 

24.  In  a  storehouse  there  is  a  bin  which  will  hold  240  bu., 
and  in  the  bin  are  183  bu.  3  pk.  of  wheat.  How  much  more 
wheat  will  the  bin  hold  ?  56  Jyii.  1  ph. 


MULTIPLICATION.  155 

SECTION   IV. 

MZrZ  TITZIC^A  TIOJV. 
190.  Ex.  Multiply  6  wk.  2  da.  8  h.  by  7. 

Explanation. — We  write  the  mul-  solution. 

tiplier  under  the  lowest  denomina-  q  y^j^^  2  da.  8  h. 

tion  of  the  multiplicand.      Then,  7 

commencing  at  the  right,  we  mul-  44  wk.  2  da.  8  h. 
tiply ;  7  times  8  h.  =  56  h.  But 
56  h.  =  2  da.  8  h.  ;  we  therefore  write  the  8  h.  in  the 
product,  and  reserve  the  2  da.  to  be  added  with  the 
days  of  the  product.  7  times  2  da.  =  14  da.,  and  14 
da.  +2  da.  =  16  da.  But  since  16  da.  =  2  wk.  2  da., 
we  write  the  2  da.  in  the  product,  reserving  the  2  wk. 
to  be  added  with  the  weeks  in  the  product.  7  times  6 
wk.  ==  42  wk.,  and  42  wk.  +  2  wk.  =  44  wk.,  which  we 
write  in  the  product.  The  result,  44  wk.  2  da.  8  h.,  is 
the  product  required. 

/■^N  l^n  OBLEMS.  ,<^\ 

Multiply  ^  T.  4:  cwt.  \^  lb.              1  cu.  yd.  14  cu.ft.  356  cu.  in. 
by  6  l_ 

3.  Multiply  4  bu.  2  pk.  7  qt.  by  9. 

4.  If  a  painter  uses  3  gal.  3  qt.  1  pt.  of  linseed-oil  in  paint- 
ing 1  lumber  wagon,  how  much  will  he  use  in  painting  5 
wagons  ?  lJi>  gal.  1  qt.  1  pt. 

5.  If  a  man  can  cradle  an  acre  of  wheat  in  3  h.  20  min., 
how  long  will  he  be  in  cutting  7  acres  ?  23  h.  20  min. 

6.  If  the  rate  of  speed  of  a  railroad  train  is  28  mi.  216  rd.  an 
hour,  how  far  will  it  run  in  14  hours  ?  JfOl  mi.  IJ^Jf.  rd. 

7.  What  is  the  weight  of  50  bales  of  cotton,  each  weighing 
4  cwt.  96  lb.  ?  12  T.  8  cwt. 

8.  How  many  bushels  of  wheat  will  a  field  of  9  acres  yield, 
at  an  average  of  14  bu.  3  pk.  4  qt.  an  acre  ?    133  hi.  3  ph  4  ^• 

9.  How  much  wood  can  a  team  draw  at  18  loads,  if  they  draw 
1  cd.  2  cd.  ft.  12  cu.  ft.  at  each  load  ?      24,  al  1  cd.ft.  8  cu.ft. 


156  COMPOUND     NUMBERS. 

191  •  Hule  for  Muttipllcation  ofConipou7idJVunibers. 

I.  Write  the  multiplier  under  the  lowest  denomination 
of  the  multiplicand. 

II.  Multiply  each  denomination  of  the  multiplicand,  in 
order,  by  the  multiplier,  as  in  integers  ;  and  when  the  prod- 
uct is  less  than  1  of  the  next  higher  denomination,  write 
it  under  the  denomination  multiplied. 

in.  When  any  product  is  equal  to,  or  greater  than,  1 
of  the  next  higher  denomination,  reduce  it  to  that  higher 
denomination,  write  the  7^emainder  under  the  denomina- 
tion multiplied,  and  add  the  quotient  with  the  next  higher 
denomination  in  the  final  result. 

Pit  OBJLJEMS. 

10.  Multiply  12  A.  84  sq.  rd.  by  27.  338  A.  28  sq.  rd. 

11.  If  a  farmer  uses  1  bu.  3  pk.  2  qt.  of  seed- wheat  to  the 
acre,  how  much  will  he  use  in  seeding  15  acres  ?      27  bu.  6  qt. 

12.  How  much  cider  will  it  take  to  fill  8  demijohns,  each 
holding  3  gal.  2  qt.  1  pt.  ?  29  gal. 

13.  A  publisher  uses  2  rm  7  quires  12  sheets  for  each  num- 
ber of  a  weekly  newspaper.  How  much  paper  does  he  use  in 
a  year  ?  123  rm.  10  quires. 

14.  If  the  water  of  a  river  flows  at  the  rate  of  3  mi.  280  rd. 
an  hour,  how  far  will  a  log  float  in  219  hours  ?     SJiS  mi.  200  rd. 

15.  If  33  cd.  7  cd.  ft.  of  wood  make  1  canal-boat  load,  how 
much  wood  will  make  19  loads  ?  6JiS  cd.  5  cd.ft. 

16.  A  farm  hand  can  plow  an  acre  of  corn  in  4  h.  15  min. 
How  long  will  it  take  him  to  plow  25  acres,  if  he  works  10 
hours  a  day  ?  10  da.  6  h.  15  min. 

17.  How  much  land  is  there  in  24  village  lots,  each  5  rods 
front  and  7  rods  deep  ?  5  A.  Ji,0  sq.  rd. 

18.  A  teamster  drew  32  loads  of  freight,  each  load  weighing 
1  T.  2  cwt.  25  lb.    How  much  freight  did  he  draw  ? 

35  T.  12  cwt. 

19.  If  a  manufacturer  makes  15  gro.  4  doz.  9  clothes  pins 
each  day,  how  many  does  he  make  in  the  308  working-days 
of  the  year?  J^,7Jfl  gro.  11  doz. 


DIVISION.  157 

SECTION   V. 

DITISIOjY, 

192i  Ex.  Divide  16  rm.  9  quires  14  sheets  of  paper 
into  5  equal  parts. 

Explanation.— We  write  solution. 

the  dividend  and  divisor,      16  rm.  9  quires  14  wheels  \  5 
and  commence  at  the  left        3  rm.  5  quires  22  sheets 
of  the  dividend  to  divide, 

as  in  integers.  One  fifth  of  16  rm.  is  3  rm.  mth  a  re- 
mainder of  1  rm.  Writing  the  3  rm.  in  the  quotient, 
we  reduce  the  1  rm.  remainder  to  quires,  and  to  it  add 
the  9  quires,  making  29  quires.  One  fifth  of  29  quires  is 
5  quires  with  a  remainder  of  4  quires.  Writing  the  5 
quires  in  the  quotient,  we  reduce  the  4  quires  remain- 
der to  sheets,  and  to  it  add  the  14  sheets,  making  110 
sheets.  One  fifth  of  110  sheets  is  22  sheets,  which  we 
write  in  the  quotient.  The  result,  3  rm.  5  quires  22 
sheets,  is  the  quotient  required. 

FROBL  EMS. 

Find  the  quotient  in  problems  1,  3,  3. 

(1)  (2)  (3) 

53  ha.  3  pi.  4  ^.  I  9     300  gro.  4  doz.  j  16      401  cd.  5  cd.ft.  I  17 

4.  A  ship  sailed  59  mi.  30  rd.  in  7  hours.     What  was  her 
average  hourly  distance  ?  8  mi.  lJf.0  rd. 

5.  A  farmer  put  385  gal.  3  qt.  1  pt.  of  cider  into  9  casks. 
How  much  cider  did  he  put  into  each  ?        Jf2  gal.  3  qt.  1  pt. 

6.  If  a  glazier  can  set  the  glass  for  8  windows  in  10  h.  40 
min.,  how  long  will  it  take  him  to  set  the  glass  for  1  window  ? 

7.  A  teamster  fed  to  his  horses  67  bu.  3  pk.  of  oats  in  30 
days.    How  many  oats  did  he  feed  each  day  ?       21m.  1  pic. 

8.  If  a  locomotive  bums  13  cd.  6  cd.  ft.  4  cu.  ft.  of  wood  in 
making  13  trips,  how  much  does  it  bum  in  making  1  trip  ? 


158  COMPOUND     NUMBERS. 

193.  ^ule  for  division  of  Compound  JVumbers. 

L   Write  the  dividend  and  divisor,  as  in  integers. 

n.  Divide  the  highest  given  denomination,  as  in  in- 
tegers, and  write  the  result  as  the  corresponding  denomina- 
tion in  the  quotient. 

ni.  Eeduce  the  remainder  to  the  next  lower  denomina- 
tion, add  to  the  result  the  number  given  of  this  lower 
denomination,  and  divide  the  same  as  before. 

IV.  Proceed  in  the  same  manner  until  all  the  denomina- 
tions of  the  dividend  are  divided. 

PM  OB  JO  JEMS. 

9.  From  15  acres  of  meadow  a  farmer  cut  28  T.  13  cwt.  75  lb. 
of  hay.    What  was  the  yield  per  acre  ?      1  T.  18  cwt.  25  lb. 

10.  He  harvested  376  bu.  3  pk.  4  qt.  of  barley  from  12  acres. 
What  was  the  yield  of  barley  per  acre  ?        311m.  1  ph  5  qt. 

11.  A  workman  laid  64  rd.  3  yd.  1  ft.  of  stone-wall  in  26 
days.     How  much  did  he  lay  each  day  ?        2  rd.  2  yd.  2  ft. 

12.  How  long  will  it  take  a  cooper  to  make  1  flour  barrel, 
if  he  can  make  8  in  10  hours  ?  1  h.  15  min. 

13.  A  bridge  pier  containing  448  cu.  yd.  of  stone  was  built 
in  36  days.  What  was  the  average  amount  of  stone  laid 
daily  ?  12  cu.  yd.  12  cu.ft. 

14.  Divide  69  T.  3  cwt.  29  lb.  8  oz.,  by  19. 

Quotient^  3  T.  12  cwt.  80  lb.  8  oz. 

15.  A  cook  uses  12  doz.  eggs  in  18  days.  How  many  eggs 
does  she  use  each  day  ?  8. 

16.  An  ink  manufacturer  put  up  3  gal.  2  qt.  1  pt.  2  gi.  of 
ink  in  59  bottles.     How  much  did  each  bottle  contain  ?    2  gi. 

17.  If  1  rm.  10  quires  of  paper  are  used  in  making  10  doz. 
writing-books,  how  many  sheets  are  used  in  making  1  doz. 
books  ?     How  many  in  making  1  book  ?      6  sheets  in  1  booh 

18.  If  12  men  can  chop  132  cords  of  wood  in  4  days,  how 
many  cords  can  1  man  chop  in  1  day  ?  2  cd.  6  cd.  ft. 

19.  If  7  men  can  mow  26  A.  40  sq.  rd.  of  grass  in  10  hours, 
how  much  can  1  man  mow  in  1  hour  ?  60  sq.  rd. 


REVIEW    PROBLEMS.  159 

SECTION    VI. 

"P^RO^SLBMS   IJ\r  COM'POU'J\r^   JVZrM:SB'RS. 

1.  How  many  bales,  each  weighing  250  lb.,  will  7  T.  5  cwt. 
of  hay  make  ?  ^^• 

2.  At  $3  a  rod,  how  much  will  it  cost  to  build  a  fence 
around  a  lot  8  rods  long  and  5  rods  wide  ?  $'78. 

3.  A  man  traveled  by  stage  78  miles  in  16  hours.  How  far 
did  he  travel  in  an  hour  ?  Jt  mi.  280  rd. 

4.  How  much  will  3  T.  5  cwt.  56  lb.  of  iron  castings  cost, 
at  7  cents  a  pound  ?  $J^58.92. 

5.  How  many  cu.  yd.  of  stone  are  there  in  an  abutment 
56  ft.  long,  8  ft.  wide,  and  15  ft.  high  ?      2J^8  cu.  yd.  2J^  cu.ft. 

6.  How  many  gallons  of  wine  will  25  doz.  quart  bottles 
hold?  75. 

7  Five  wood-cutters  worked  together  during  the  winter. 
The  first  chopped  118  cd.  4  cd.  ft.,  the  second  109  cd.  2  cd.  ft. 
8  cu.  ft.,  the  third  106  cd.  5  cd.  ft,  the  fourth  98  cd.  3  cd.  ft. 
8  cu.  ft.,  and  the  fifth  91  cd.  7  cd.  ft.  How  much  wood  did 
they  chop  ?  52 J^  cd.  6  cd.ft. 

8.  From  Dec.  30  to  Aug  1  of  the  following  year,  how  many 
months  and  days  ?  1  mo.  2  da. 

9.  How  many  minutes  from  Aug.  18,  at  noon,  to  Oct.  9,  at 
noon?  7J!i,,880  min. 

10.  How  much  cider  can  be  made  from  100  bu.  of  apples,  if 
3  gal.  1  qt.  1  pt.  can  be  made  from  1  bu.  ?         337  gal.  2  qt. 

11.  From  a  barrel  which  contained  42  gal.  2  qt.  of  syrup, 
7  gal.  2  qt.  were  drawn  one  day,  3  gal.  1  qt.  1  pt.  the  next, 
and  4  gal.  3  qt.  the  third  ?  How  much  syrup  remained  in  the 
barrel  ?  26  gal.  3  qt.  1  pt. 

12.  A  railroad  company  has  a  pile  of  wood  576  ft.  long, 
25  ft.  wide,  and  18  ft.  high.  How  much  wood  is  there  in 
the  pile?  2,025  cd. 

13.  In  17  gal.  1  pt.,  how  many  gills  ?  5J^8. 

14.  At  $.625  a  cu.  yd.,  how  much  will  it  cost  to  dig  a  cellar 
27  ft.  long,  19  ft.  wide,  and  7  ft.,  deep  ?  $83,125. 


160  COMPOUND     NUMBERS. 

15.  A  way-freight  car  was  loaded  with  3  T  3  cwt.  48  lb.  of 
groceries,  3  T.  19  cwt.  40  lb.  of  hardware,  1  T.  1  cwt.  94  lb, 
of  furniture,  and  18  cwt.  64  lb.  of  dry  goods.  How  much 
freight  was  in  the  car  ?  9  T.  3  cwt.  lf,6  Tb. 

16.  A  druggist  bought  9  casks,  each  containing  20  gal.  3  qt. 
1  pt.  of  brandy.     How  much  did  they  all  contain  ? 

17.  How  much  seed-corn  will  be  required  for  5,000  pint 
papers  ?  75  lu.  4  qt. 

18.  How  many  cu.  in.  are  3  cu.  yd.  18  cu.  ft.  334  cu.  in.  ? 

19.  If  2  show-bills  can  be  printed  on  1  sheet,  how  much 
paper  will  be  required  for  2,400  bills  ?  2  rm.  10  quires. 

20.  At  $35  an  acre,  what  will  be  the  cost  of  a  piece  of  land 
140  rd.  long,  and  112  rd.  wide  ?  $3,JjS0. 

21.  How  many  3-pint  bottles  will  32  gal.  2  qt.  1  pt.  of  cider 
fill  ?  87. 

22.  If  it  is  18  feet  around  the  hind  wheel  of  a  carriage,  how 
many  times  will  the  wheel  turn  over  in  runnmg  6  mi.  174  rd. 
3  yd.?  1,920. 

23.  A  certain  room  is  22  ft.  long,  18  ft.  wide,  and  12  ft.  high. 
How  many  sq.  yd.  are  there  in  the  ceiling  ?  W 

24.  How  many  square  yards  are  there  in  the  four  sides  of 
the  same  room  ?  106  sq.  yd.  6  sq.ft. 

25.  A  grocer  bought  6  doz.  brooms  for  $19.34,  and  retailed 
them  at  40  cents  apiece.     How  much  did  he  gain  ? 

26.  In  making  a  road  250  rods  long,  5  cu  yd.  3  cu.  ft.  of 
gravel  were  used  to  the  rod.     How  much  gravel  was  used  ? 

1,277  cu.  yd.  21  cu.ft. 

27.  If  a  manufacturer  makes  1,000  lead-pencils  in  a  day, 
how  many  gross  will  he  make  in  26  days  ?    180  gro.  6  doz.  8. 

28.  A  man  bought  7  acres  of  land,  at  $450  an  acre,  and 
sold  it  in  building  lots,  each  10  rd.  long  and  4  rd.  wide,  at 
$150  apiece.     How  much  did  he  gain  ?  $1,050. 

29.  In  1  liquid  gallon  there  are  231  cu.  in.  What  is  the 
capacity  in  gallons  of  a  cistern  7  ft.  long,  5.5  ft.  wide,  and  9 
ft.deep?  ^  ^^^^^• 

30.  How  long  will  3  cwt.  of  sugar  last  a  family,  if  they  use 
lib.  a  day?  42  wh  6  da. 


REVIEW    PROBLEMS.  161 

31.  What  will  be  the  cost  of  a  pile  of  stone  30  ft.  long,  8  ft 
wide,  and  4  ft.  high,  at  $6  a  cord  ?  $45. 

32.  How  many  miles  will  a  locomotive  run  in  4  hours,  run 
ning  at  the  rate  of  124  rods  in  a  minute  ?  93. 

33.  Last  year,  I  sold  from  my  garden  5  bu.  1  qt.  of  cherries, 
dried  1  bu.  3  pk.  1  qt.,  put  up  in  cans  3  pk.  5  qt.,  and  3  pk. 
2  qt.  were  eaten  in  my  family.  How  many  cherries  grew  in 
my  garden  ?  8  hi.  2  ph.  1  qi. 

34.  These  cherries  grew  upon  7  trees.  What  was  the  aver- 
age yield  per  tree  ?  l'bu.7qt.' 

35.  How  many  tons  of  hay  will  a  span  of  horses  eat  in  15 
weeks,  if  they  eat  45  lb.  a  day  ?  2  T.  7  cwt.  25  lb. 

36.  How  many  oats  will  it  take  to  last  them  the  same  time, 
if  they  eat  24  qt.  a  day  ?  78  du.  3  pi. 

37.  If  a  family  use  3  bu.  1  pk.  of  potatoes  each  month,  how 
much  will  a  year's  supply  cost  them,  at  $.5625  a  bushel  ? 

38.  If  a  housekeeper  uses  a  half-pint  of  molasses  each  day, 
how  long  will  20  gallons  last  her  ?  Jf5  wTc.  5  da. 

39.  A  plumber  has  a  coil  of  lead  pipe  34  ft.  long,  which 
weighs  189  lb.  2  oz.    How  much  does  1  ft.  of  the  jDipe  weigh  ? 

40.  How  much  will  15  ft.  of  the  same  pipe  weigh  ? 

41.  If  I  deposit  $4.50  in  a  savings-bank  eveiy  week,  and 
draw  out  $12.50  each  month,  how  much  will  I  have  on 
deposit  at  the  end  of  the  year  ?  $8Jf. 

42.  How  many  cords  of  stone  will  it  take  to  build  a  stone 
fence  76  ft.  long,  4  ft.  high,  and  2  ft.  thick  ?  ^.75  cd, 

43.  One  day  a  carman  drew  14  T.  18  cwt.  52  lb.  of  freight, 
at  17  equal  loads.     How  much  did  he  draw  at  each  load  ? 

44.  A  grocer  bought  four  hogsheads  of  molasses,  which  con- 
tamed  118  gal.  3  qt.,  123  gal.  2  qt.,  109  gal.,  and  122  gal.  1  qt. 
How  much  molasses  did  he  buy  ? 

45.  From  the  first  hogshead  he  sold  49  gal.,  from  the  sec- 
ond 68  gal.  3  qt.,  from  the  third  39  gal.  1  qt.,  and  from  the 
fourth  54  gal.  2  qt.     How  much  molasses  did  he  sell  ? 

46.  How  much  molasses  was  left  in  each  hogshead  ? 

In  M  and  3d,  69  gal.  3  qt. ;  in  2d,  54  gal.  3  qt. ;  in  4th,  67  gal.  3  qt. 

47.  How  much  molasses  had  he  on  hand  ?  262  gal. 

N 


CHAPTER  IV. 
FRACTIONS. 

SECTION  I. 
ij\ri>iTCTioj\r  Ajv^  j\roTATioj\r, 

194i  When  an  apple  is  divided  into  2  equal  parts,  1 
of  the  parts  is  1  half.  When  a  pear  or  any  thing  is 
divided  into  3  equal  parts,  1  of  the  parts  is  1  third,  and 
2  of  the  parts  are  2  thirds.  When  a  thing  is  divided 
into  4  equal  parts,  the  parts  are  fourths. 

1  half  is  written  \ ;  1  third,  \  ;  2  thirds,  § ;  1  fourth,  \ ; 
2  fourths,  I;  3  fourths,  |. 

When  a  thing  or  a  number  is  divided  into  5  equal 
parts,  the  parts  are  fifths  ;  when  into  6  equal  parts, 
they  are  sixths ;  when  into  7  equal  parts,  they  are 
sevenths ;  and  when  into  8  equal  parts,  they  are 
eighths.  Fifths,  sixths,  sevenths,  and  eighths  are  writ- 
ten thus  : 


1  fifth,  \, 

2  sixths,  1, 

1  seventh,    ^, 

4  eighths,  f, 

2  fifths,  1, 

4  sixths,  f , 

3  sevenths,  5^ 

5  eighths,  |, 

4  fifths,  |. 

5  sixths,  %. 

6  sevenths,  %. 

7  eighths,  \, 

INDUCTION     AND      NOTATION.  1G3 

195.  A  number  that  represents  one  or  more  of  the 
equal  parts  into  which  a  thing  is  divided  is  a  Fraction. 

196.  The  two  numbers  that  are  used  in  writing  a 
fraction  are  the  Terms.  Thus  the  terms  of  the  fraction 
§  are  5  and  6.  The  6  shows  that  a  whole  thing  is 
divided  into  6  equal  parts,  and  the  5  represents  5  of 
these  parts. 

197.  The  term  that  expresses  the  number  of  equal 
parts  into  which  a  whole  one  is  divided  is  written 
below  a  horizontal  line,  and  is  the  Denominator  ;  and 

198.  The  term  that  represents  the  number  of  these 
equal  parts  is  written  above  the  Hne,  and  is  the 
Numerator.  Thus,  in  the  fraction  ^,  4  and  7  are  the 
terms  ;  4  is  the  numerator,  and  7  is  the  denominator. 

199.  When  the  numerator  is  less  than  the  denom- 
inator, the  fraction  is  less  than  1 ;  as,  |,  f ,  |,  j^,  -f\. 

When  the  numerator  and  denominator  are  equal, 
the  value  of  the  fraction  is  1 ;  as,  f  =  1,  |  =  1,  j|  =  1, 

15   _  1      24  _  1 

15  —  ^}    24  —  •*•• 

When  the  numerator  is  greater  than  the  denomina- 
tor, the  value  of  the  fraction  is  greater  than  1 ;  as,  |,  |, 

12     JJl    15      2  7 
~o'>     9  »  «d2'    2  0* 

200.  A  fraction  whose  value  is  less  than  1  is  a 
Proper  Fraction  ;  as,  |,  |,  j\,  Jf ,  3^. 

201.  A  fraction  whose  value  is  equal  to  or  greater 
than  1  is  an  Improper  Fraction  ;  as,  |,  |,  f ,  -y-,  J/-,  if. 

202.  Fractions  whose  denominators  are  alike  are 
Similar  Fractions  ;  as,  |,  |,  and  |  ;  |,  |,  J,  and  |. 

203.  Fractions  whose  denominators  are  unlike  are 

Dissimilar  Fractions  ;  as,  |,  |. 

204.  A  number  composed  of  an  integer  and  a  frac- 
tion is  a  Mixed  Number;  as,  4f,  31|.     (See  119.) 


164 


FRACTIONS. 


3x| 


f,  and  4  X  I  =  I. 


205.  Two  of 

the  8  equal 
parts  of  this 
cake,  or  |,  are  2 
times  as  much 
as  1  of  the 
parts,  or  I;  and 

1  are  2  times 
as  much  as  f . 
Hence,  2  x  |  = 
I,  and  2  X  f 
=  |.     So,  also,  3  X  I  = 

If  we  divide  §  of  the  cake  into  2  equal  parts,  i  part 
will  be  I ;  if  we  divide  |  of  it  into  2  equal  parts,  each  of 
the  2  equal  parts  will  be  f .  So,  also,  |^2  =  |,  f-^2 
—  7»  3    •   ^  —  3>  TT   •   ^  —  TT*     Jj-cnce, 

I.  A  fraction  may  he  multiplied  by  multiplying  its 
numerator. 

II.  A  fraction  may  he  divided  hy  dividing  its  numerator, 

206.  If  we  divide  a  whole  cake  into  2  equal  parts, 
each  part  wiU  be  | ;  if  we  divide  ^  of  it  into  2  equal 
parts,  each  part  will  be  |  ;  and  if  we  divide  |  of  it  into 

2  equal  parts,  each  part  will  be  |.     That  is,  1  -^  2  =  ^, 
I  —  2  =  I,  and  ^  -^  2  =  i. 

Again,  2  of  the  eighths  put  together  are  |,  2  of  the 
fourths  together  are  ^,  and  the  2  halves  are  the  whole 
X  1  =  4, 2  X  i  =  i,  and  2  X  A  =  1. 


caJce,  or  1.   That  is,  2 


I.  A  fraction  may  be  divided  hy  multiplying  its  denom- 
inator. 

II.  A  fraction  may  he  multiplied  hy  dividing  its  denom- 
inator. 

207.  If  we  divide  ^  of  a  melon  into  2  equal  parts,  we 
shall  have  |  of  a  melon  ;  and  if  we  divide  |  of  a  melon 


INDUCTION   AND   NOTATION 


16i 


into  2  equal 
parts,  we  shall 
have  §  of  a  mel- 
on.    That  is,  ^ 


and  I 


But  the  ^  may 
be  changed  to 
f ,  and  the  i  to 
f ,  by  multiply- 
ing both  terms 
of  each  frac- 
tion by  2.     Thus, 


1X2   —  2 

3  X  2   —  4> 


and 


1   X  2 
4X2 


Again,  |  of  the  melon  are  together  equal  to  |  of  it? 
and  I  of  a  melon  are  together  equal  to  ^  of  it.  That 
is,  1  =  1,  and  j  =  J.  But  the  f  may  be  changed  to 
I,  and  the  f  to  |,  by  dividing  both  terms  of  each  frac- 
tion by  2.     Thus,  §  i  i  =  |,  and  1 1: 1  =  i-     Hence, 

I.  The  value  of  a  fraction  is  not  changed  by  multiplying 
both  terms  by  the  same  number. 

n.  The  value  of  the  fraction  is  not  changed  by  dividing 
both  terms  by  the  same  number. 

208.  All  operations  in  fractions  are  based  upon  the 
following 

General  Principles  of  JFraclions, 

I.  Afrajction  is  multiplied, 

1.  By  multiplying  its  numerator  ;  or, 

2.  By  dividing  its  denominator. 
n.  A  fraction  is  divided, 

1.  By  dividing  its  numerator  ;  or, 

2.  By  multiplying  its  denominator. 
m.   TJie  value  of  a  fraction  is  not  changed, 

1.  By  multiplying  both  terms  by  the  same  number;  or, 

2.  By  dividing  both  terms  by  the  same  number. 

(See  Manual,  page  219.) 


166  FRACTIONS. 

SECTION  II. 

^B^  ZTC  TlOJSr. 

C-A.se    I, 
Fractions  to  Lo^vest  Terms. 

209.  When  tlie  terms  of  a  fraction  can  not  both  be 
exactly  divided  by  any  integer  greater  than  1,  the  frac- 
tion is  in  its  lowest  terms.  Thus,  f  is  in  its  lowest 
terms,  because  no  integer  greater  than  1  will  exactly 
divide  both  5  and  7. 

210.  To  reduce  a  fraction  to  its  lowest  terms  is  to 
change  its  numerator  and  denominator  to  the  smallest 
numbers  possible,  without  changing  the  value  of  the 
fraction.     Thus,  ^  —  \=^\. 

Ex.  Keduce  jf  to  its  lowest  terms. 

Explanation. — We  reduce  ||  to  lower     first  solution. 
terms  by  dividing  both  numerator  and     i|  =  |  =  | 
denominator  by  2,    (jf  =  -|) ;    and  the 
fraction,    f ,    thus    obtained,   we    reduce  second  solution. 
to    still   lower   terms   by  dividing  both         i|  =  | 
terms  by  3,  (f  =  |,)  as  shown  in  the  First 
Solution.     (See  Prin.  HE.,  2.)     Or  we  can  reduce  jf  to 
its  lowest  terms   at  one   operation  by  dividing  both 
terms  by  6,  as  shown  in  the  Second  Solution. 

1.  Reduce  the  fraction  \  to  its  lowest  terms.  ^. 

2.  Reduce  /^  mi.  to  its  lowest  terms.  |-  mi. 

3.  To  what  lower  terms  can  ||  be  reduced?       J<^^  ^,  or  f. 

4.  Reduce  ^-^^\\,  and  /^  to  their  lowest  terms.      |-^  |,  j.. 

5.  In  what  lower  terms  can  lyV^  be  expressed  ? 


REDUCTION.  167 

6.  Reduce  f  |,  \^,  and  ||  to  their  lowest  terms. 

7.  Redu6e  the  fractions  f  f  and  \y  to  their  lowest  teims. 

h  V-. 

8.  What  are  the  lowest  terms  of  the  fractions  {■q%,  j%,  t^, 
1^,  rVa,  and  iff  ?  ff,  |,  f,  |,  Iv,  f ." 


C-A.se    II. 
Fractions  to  Given  Denominators. 

211.  "We  have  already  seen  that  the  value  of  a  frac- 
tion is  not  changed  by  multiplying  both  terms  by  the 
same  integer.  Thus,  |^  |  ==  J,  f  >^  f  _  _6^^  |  x  4  _,  ^2^ 
I  X  i  =  If,  and  so  on. 

Ex.  Eeduce  |  to  thirty-sixths. 

Explanation. — ^To  reduce  2  ninths  to  solution. 

thirty-sixths,  we  must  multiply  both  terms  36  [  9 

by  such  an  integer  as  will  give  36  for  the  4 
new  denominartor.     We  find  this  integer 

by  dividing  36  by  9.     Multiplying  both  I  x  4  =  3% 
terms  of  |  by  4,  the  integer  thus  found, 
we  have  /g.                                                Hence,  f  =  /g^. 

JPJtOBZJEMS. 

9.  Reduce  f  to  twelfths.  Z^-. 

10.  Reduce  f  to  twenty-firsts. 

11.  Reduce  |  to  eighteenths,  and  |  to  twenty-fourths,  -f^,  /j. 
13.  Reduce  |  to  sixths,  to  ninths,  and  to  fifteenths. 

A   1^  and  if 

13.  Reduce  f  to  forty-fifths,  and  j\  to  thirty-thirds. 

14.  Reduce  f  to  fourteenths  and  to  twenty-firsts. 

15.  Reduce  f  to  fortieths  and  |  to  fortieths.  ||^  If. 

16.  Reduce  f  and  f  to  twenty-eighths. 


168  FBACTIONS. 

CASE    III. 
Dissimilar  Fractions  to  Similar  Fractions. 

212i  Ex.  1.  Beduce  |  and  J  to  similar  fractions. 

Explanation. — Thirds  can  not  be  solution. 

reduced  to  halves,  nor  halves  to  2  x  2  _  4 

thirds.     But  since  2  times  3  =  6, 
we  reduce  |  to  sixths  by  multiply-  ^-  ^  |  =  | 

ing  both  terms  by  2 ;  and  since  3 
times  2  =  6,  we  reduce  ^  to  sixths    Hence,  §,  ^=§,  §, 
by  multiplying  both  terms  by  3. 

Ex.  2.  Keduce  f ,  1,  and  f  to  similar  fractions. 

Explanation.— Since  3  x  4  x  7  =  84,  solution. 

we  reduce  these  fractions  to  84ths  ix4x7  =  if 
by  multiplying  both  terms  of  the  |  ^  §  x  7  _  2 1 
first  by  4  and  7,  both  terms  of  the  5X3X4_6o 
second  by  3  and  7,  and  both  terms  ^  ><  s  x  4  -  84 
of  the  third  by  3  and  4.     Hence,  f ,  |,  f  =  f f,  f |.  f f . 

213.  Hence,  to  reduce  dissimilar  to  siii^ilar  fractions, 
we 

Multiply  both  terms  of  each  fraction  by  the  denominators 
of  all  the  other  fractions. 

214.  Fractions  having  like  denominators  are  said  to 
have  a  common  denominator ;  and  reducing  dissimilar 
to  similar  fractions  is  sometimes  called  reducing  them 
to  equivalent  fractions  having  a  common  denominator. 

JPjR  oblems. 

17.  Reduce  f  and  ^  to  similar  fractions.  ^§,  ^V* 

18.  Reduce  ^  and  f  to  similar  fractions. 

19.  Reduce  f  and  ^  to  similar  fractions.  §^,   5^. 

20.  What  similar  fractions  are  equal  to  |  and  I  ? 

21.  What  similar  fractions  are  equal  to  |  and  ^g  ? 

22.  What  similar  fractions  are  equal  to  jf  and  I  ? 


REDUCTION.  169 

23.  Reduce  |  and  i\  to  similar  fractions.  §§,  ^|. 

24.  Reduc-j  i,  f,  and  f  to  similar  fractions.         ^^  ||_,  |^. 

25.  Reduce  f ,  ^,  and  f  to  similar  fractions,  ff^^  f-^-^^  ±^^. 

26.  Reduce  |,  |,  ^,  and  f  to  equivalent  fractions  having  a 
common  denominator.  //^,  //^,  //^,  //g.. 

37.  Reduce  |,  |,  f ,  and  f  to  similar  fractions. 

28.  Reduce  |,  j^2,  and  f  to  similar  fractions. 

29.  What  similar  fractions  are  equal  to  |,  ^,  and  |  ? 

30.  Reduce  i\,  ^,  f ,  and  y*j  to  similar  fractions. 

31.  Reduce  ^,  |,  and  f  to  equivalent  fractions  having  a  com- 
mon denominator.  -^^,  ||-_,  ^^, 

O-AlSE   IV. 
Improper  Fractions  to  Integers  or  Mixed  Numbers. 

15(  Ex.  1.  In  y-  how  many  ones  ? 

Explanation. — Since  every  4  solution. 

fourths  are  1,  the  number  of      12  fourths  [  4:  fourths 
times    12    fourths   contains    4      ~^ 
fourths  is  the  number  of  I's  in 
12  fourths,  and  12  fourths  con-  Hence,  ^  =  S. 

tains  4  fourths  3  times. 

Ex.  2.  In  J3O-  how  many  ones  ?  solittion. 

Tji^^^  .^,. „     -in  +i.,-^^«  «^^  10  thirds  I  3  ^/iWs 

ILxPLANATiON. — 10  tmros  con-        —  ^ 

tains  3  thirds  3  times,  with  a  ^i 

remainder  of  1  third.  (See  Manual,  p.  219.)  Hence,  ^  =  S{'. 

216.  Hence,  to  reduce  an  improper  fraction  to  an 
integer  or  a  mixed  number,  we 

Divide  the  numerator  by  the  denominator, 

PJROBZEMS, 

32.  In  -3/-  how  many  I's  ?  13. 
83.  ?f  ^  are  how  many  I's  ? 

34.  -Y"  apples  are  how  many  apples  ? 

35.  Y-  melons  are  how  many  melons  ?  <^4. 

O 


170 


FEACTIONS. 


36.  Reduce  V  to  a  mixed  number.  IS^,  or  13^, 

37.  Reduce  Y?  V)  ^^^  V/  *^  mixed  numbers. 

38.  \%^  gallons  are  liow  many  gallons  ?  <^||. 

39.  How  much  hay  in  53  bales,  each  containing  ^  T.  ? 

40.  How  many  bushels  of  peaches  in  176  baskets,  each  con^ 
taining  |  of  a  bushel  ?  S8§. 

41.  Reduce  Yi^  'IP?  ^^^  W^  to  integers  or  mixed  num- 
bers. 7^,  2SS^§-,  23. 


:]Lntegers  or  Mixed  Numbers  to  Improper  Fractions. 

217i  Ex.  1.  Keduce  5  to  fourths. 

Explanation.  —  Since  1  is  4 
fourths,  5  are  5  times  4  fourths, 
or  20  fourths. 


BOLUTION. 

4  fourths 
_5 

20  fourths 

Hence,  5  ~ 


Ex.  2.  Reduce  5|  to 
an  improper  fraction. 

Explanation. — Since  1 
is  4  fourths,  5  are  5 
times  4  fourths,  or  20 
fourths,  and  20  fourths 
+  3  fourths  are  23  fourths. 


FULL   SOLUTION. 

4:fourths 
5 


20fourths 

d  fourths 

2'Sfourths 

Hence,  5| 


COMMON  BOLUTION. 


f 


20  +  3  =  23 


(See  Manual,  page  219.) 

21 8t  Hence,  to  reduce  an  integer  or  a  mixed  number 
to  an  improper  fraction,  we* 

First  multiply  the  integer  by  the  denominator,  and,  if 
there  be  a  numerator,  add  it  to  the  product ;  then  write 
this  result  for  the  numerator,  and  the  given  denominator 
for  the  denominator  of  the  required  fraction. 


PMOBI.JSM8. 

42.  In  15  how  many  thirds  ? 

43.  In  24|  how  many  fourths  ? 

44.  What  improper  fraction  is  equal  to  17|  ? 


ADDITION.  171 

28A 


45.  Reduce  31^  to  an  improper  fraction. 

46.  Reduce  37  to  sixths,  and  11^*5  to  fifteenths. 

47.  What  improper  fraction'is  equal  to  4:-^^  ? 

48.  How  many  fifty-seconds  in  1\\  ? 

49.  12,^3  equals  how  many  thirteenths  ? 

50.  41  =  how  many  thirty-firsts  ?  '  HV- 

51.  Reduce  24^,  and  22^3  to  improper  fractions. 

52.  Reduce  5||,  214|,  and  IIS^V,  to  improper  fractions. 


SECTION  III. 

•  CASE    I. 

All  the  Parts  Fractions. 

219.  Ex.  1.  What  is  the  sum  of  tV+t3  +  t5  ai^d  t-j? 

Explanation. — Since  the  parts  boltttion. 

in  these  fractions  are  all  of  the  J^  +  ^3^  4.  _5^  j^  _2^ — j  j 
same    kind    or    denomination, 

(twelfths),  and  since  the  numerators  express  the  num- 
bers of  the  parts,  we  add  the  fractions  by  adding  their 
numerators.  1  +  3-1-5  +  2  =  11;  and  since  the  parts 
are  twelfths,  we  write  the  denominator  12  under  the  11. 
Ex.  2.  What  is  the  sum  of  |,  j,  and  |  ? 

Explanation.  —  ^^^  solhtiox. 

Fifths,  thirds,  and      J  +  ^l  =  iS  +  IB  +  l§=W-=l|| 
fourths  do  not  ex- 
press   the    same  second  solution. 
kind  of  things  or       l  +  KI  =  ''^^0^''  =  W  =  1|« 
parts,  and  hence 

they  can  not  be  directly  added  (See  20, 1).  But  they 
can  all  be  reduced  to  similar  fractions,  (sixtieths),  and 
these  similar  parts  can  be  added,  as  shown  in  Ex.  1. 


172  FRACTIONS. 

In  reducing  the  dissimilar  fractions  to  similar  ones, 
the  common  denominator  need  be  written  but  once, 
and  the  several  numerators  may  be  written  above  it,  as 
shown  in  the  Second  Solution.  (See  Manual,  page  219.) 

220.  Hence,  to  add  fractions,  we 

Reduce  all  dissimilar  to  similar  fractions,  add  the  nu- 
merators, and  under  the  sum  write  the  common  denom- 
inator. 

JPM  OBJ0EM8. 

1.  What  is  the  sum  of  f  and  ^  ?  §f, 

2.  Add  I  and  I 

3.  Add  I  and  |.    Add  f%  and  j-%.  f|;  /^. 

4.  If  a  family  bum  ^  T.  of  coal  one  month,  and  |  T.  the 
next,  how  much  do  they  bum  in  the  two  montks  ? 

5.  What  is  the  sum  of  f ,  ^,  and  |  ?  ±f§,  or  1//^. 

6.  A  merchant  sold  |  bu.  of  clover  seed  to  one  farmer,  i  bu. 
to  another,  and  |  bu.  to  a  third.  How  much  clover  seed  did 
he  sell  ?  ^//-,  or  i|  hu. 

7.  A  teamster  drew  in  three  loads,  |  cd.  of  wood,  f|  cd., 
and  \l  cd.  How  many  cords  of  wood  did  he  draw  in  the 
three  loads  ?  f^f§,  or  ^|f  cd. 

8.  What  is  the  sum  of  |  and  f  ? 

9.  Add  -y,,  f,  and  f.  f§j,  or  2/^. 

10.  A  market  gardener  has  f  of  an  acre  of  blackberries,  ^ 
of  an  acre  of  raspberries,  and  f  of  an  acre  of  strawberries. 
How  many  acres  of  berries  has  he  ? 

11.  Keduce  ^,  f,  and  f  to  twelfths,  and  find  their  sum. 

12.  Reduce  ^  da.  ^  da.  |  da.  and  |  da.  to  twenty-fourths  of 
a  day,  and  add  them.  §^,  or  l^j  dwys. 

13.  Reduce  %\,  $f ,  $,-\,  and  IgV  to  himdredths  of  a  dollar, 
and  find  their  sum.  $l-f^Q. 

14.  A  sewing  girl  paid  %\  for  a  thimble,  $yV  for  needles, 
%%  for  silk  braid,  IfW  ^^^  sewing  silk,  and  %^-^  for  a  spool  of 
cotton.  How  many  himdredths  of  a  dollar  did  each  article 
cost  ?    How  much  money  did  she  pay  out  ?  ^l^U' 

% 


ADDITION.  173 

CA.SE     II. 
All  or  Some  of  the  Parts  Mixed  Numbers. 

221.  Ex.  What  is  the  sum  of  2|,  5^,  7,  and  ^  ? 
Explanation. — We  write  the  parts  with        boltjtion. 
the  integers  in  the  same  column,  and  re-      2 1  =  2f  ^ 
duce  the  dissimilar  fractions  to    similar      5|  =  5Jg 
ones.     Fifths,  halves,  and  thirds  can  be      7=7 
reduced  to  thirtieths.      Since  |  =  |4,  2|      4|  =  4f  g 
must  equal  2f  ^.     So,  also,  5 J  =  5|f,  and  ^^'9 

4f  =  4§g.     Adding  the  fractions,  we  have  ^^ 

|§,  or  If  J.     We  write  the  f  §  in  the  result,  and  add  the 

1  with  the  given  integers.     The  sum  of  aU  the  integers, 
19,  written  before  the  f  §,  gives  19§§,  the  required  sum. 

PROBLEMS, 

15.  Add  2f  and  4f .  ^  7/y. 

16.  What  is  the  sum  of  ^  and  3f  ?  * 

17.  A  copper-smith  used  5f  bu.  of  charcoal  one  month,  and 
6|  bu.  the  next.     How  many  bushels  did  he  use  in  the  two 

months  ?     (Fourths  may  be  reduced  to  eighths.)  12 f  du. 

18.  A  farmer  has  85|  A.  of  cleared  land,  and  47f  A.  of  wood- 
land.   How  many  acres  are  in  his  farm  ? 

19.  What  is  the  sum  of  241  j%,  4f ,  and  1§  ?  24-7^. 

20.  Add  8|,  6^,  and  27f . 

21.  Mr.  Wood's  farm  is  If  mi.  long,  and  f  mi.  wide.    What 
is  the  distance  around  it  ?  3^§  mi. 

22.  A  teamster  drew  two  loads  of  straw,  one  weighing  l^'V 
T.,  and  the  other  H  T.    How  much  straw  in  both  loads  ? 

23.  What  is  the  sum  of  $4f ,  $61,  $23,  and  $if  ? 

24.  What  is  the  sum  of  l/o,  3^,  5f ,  and  lOf  ?         21^§i. 

25.  A  fruit  dealer  bought  l^f  bu.  of  walnuts  of  one  boy,  and 

2  ^  bu.  of  another.   How  many  walnuts  did  he  buy  ?     S§§  bu. 

26.  Add  3i,  14,  5t^3,  I  41.  27^- 

27.  Add  19f ,  j\,  I,  651,  and  23.  109//^. 


174  FRACTIONS. 

SECTION   IV. 

S  U:B  T^A  C  TIOJV. 

OA.SE    I. 
Both  Numbers  Fractions. 
222.  Ex.  1.  From  \l  subtract  j^. 
Explanation.  —  Since  tlie  fractions  solution. 

are  similar,  we  subtract  tbe  numerator      j  J  —  j^^  —  -^^ 
of  the  less  fraction  from  that  of  the 
greater.      11  —  7  =  4  ;  and  since  the  parts  are  all  of 
the  same  kind  or  denomination,  (fifteenths),  we  write 
the  denominator,  15,  under  the  4. 

Ex.  2.  From  i  subtract  |. 

Explanation. — Since  fifths  first  solution. 

and  thirds  do  not  express  the       5~3==T5"~T5— To 
same  kind  of  things  or  parts,  g^^oxD  solution, 

(See   33,  I),  we  reduce  the         4  _  2  _.  12  -  ro  _  _2_ 
given    fractions    to    similar         53  lo  lo 

fractions,  (fifteenths),  and  then  subtract  the  less  from 
the  greater  in  the  same  manner  as  shown  in  Ex!  1. 

In  reducing  the  dissimilar  fractions  to  similar  ones, 
the  common  denominator  need  be  written  but  once. 

223.  Hence,  to  subtract  fractions,  we 

Reduce  all  dissimilar  to  similar  fractions,  subtract  the 
less  numerator  from  the  greater,  and  under  the  difference 
write  the  common  denominator. 

PB  OBZEM8, 

1.  From  i  subtract  f^g.  -^j^. 

2.  From  |  subtract  f.  ^J. 

3.  What  is  the  diflference  between  }f  and  \  ? 

4.  What  is  the  diflference  between  $|  and  $/g  ?  ${-, 


SUBTRACTION.  175 

5.  From  a  jug  that  contained  |  of  a  gallon  of  boiled  cider, 
a  woman  used  5  of  a  gallon.  How  much  cider  was  left  in  the 
jug?  §gal 

6.  One  day  A  worked  -{'^  of  the  day,  and  B  ^  of  the  day. 
Which  worked  the  longer  ?    How  much  the  longer  ? 

7.  A  housekeeper  bought  i  pk.  of  cranberries,  and  used  I 
pk.  the  same  day.     How  many  had  she  left  ?  ^^  ph 

8.  From  f  subtract  |.     From  ^  subtract  ^^ 

9.  Charles  lives  f|  mi.  from  the  schoolhouse,  and  John  ? 
mi.  Which  lives  the  greater  distance  from  the  schoolhouse  ? 
How  much  the  greater  ?  Charles,  /^  mi. 

10.  The  snow  was  -^^  ft.  deep  one  night,  and  the  next  morn- 
ing it  was  I  ft.  deep.  What  depth  of  snow  had  fallen  during 
the  night?  ^ft. 

11.  From  y\  subtract  f. 

12.  From  y\  subtract  ^.  ^^. 

13.  If  I  pour  y\  qt.  of  wine  from  a  bottle  containing  I  qt., 
hpw  much  wine  will  be  left  in  the  bottle  ?  -/g-  qt. 

14.  From  f  |  cd.  of  wood  a  teamster  took  {'^  cd.  How  much 
wood  remained  ? 

15.  A  farmer  bought  ^l  T.  of  plaster,  and  sowed  yV  T.  on 
his  clover  lot.     What  part  of  a  ton  had  he  left  ?  §§  T. 

16.  What  is  the  difference  between  f§  mi.  and  f  mi.  ? 

/s-  mi. 

C-A.se     II. 
The  Minuend  a  Mixed  Number  or  an  Integer. 

224.  Ex.  1.  From  5|  subtract  2^. 
Explanation. — ^We  write  the  subtrahend        solution. 
under  the  minuend,  and  reduce  the  frac-       5|  =  S/j 
tional  parts  to  similar  fractions.     We  then       2|  ==  2j% 
subtract  the  fractional  part  of  the  subtra-  ^5 

hend  from  that  of  the  minuend,  and  the  *' 

integer  of  the  subtrahend  from  the  integer  of  the  min 
uend.     The  result,  Sjh,  is  the  required  difference. 


1  -2 


176  FRACTIONS. 

Ex.  2.  From  7|  subtract  4|. 

Explanation. — After  reducing  the  boltttion. 

fractional  parts  of  the  given  numbers  7|  =  7^|  =  6f  g 
to  similar  fractions,  we  find  that  the  4|  =  4:|§  =  4|g 
fraction  of  the  subtrahend  is  greater  W^ 

than  that  of    the   minuend.      "We  ^^ 

therefore  take  1  of  the  7,  and  unite  its  value  (|J)  with 
the  |§,  thus  changing  the  minuend  to  6f  §.  From  this 
we  subtract  4|§,  in  the  manner  shown  in  Ex.  1. 

Ex.  3.  From  15  subtract  4f .         solution. 

Explanation. — ^Before  subtracting,  we  15  =  14| 
reduce  1  of  the  15  to  sevenths,  thus  4f  =  4^ 
changing  the  minuend  to  14|.  j^T 

PItOBIj:EM8. 

17.  From  7|  subtract  3|.  4^, 

18.  From  7^  subtract  3f.  S§§, 

19.  From  8f  subtract  fi.  7§§.M 

20.  A  merchant  bought  a  cheese  which  weighed  BOf  lb.,  and 
sold  23|  lb.  of  it.     How  much  cheese  had  he  left  ?     57§-  lb. 

21.  A  piece  of  cloth  that  measured  43^  yd.  before  it  was 
dressed,  shrank  2f  yd  in  fulling.  How  many  yards  did  it 
then  contain  ?  ^^H- 

22.  From  19y«y  subtract  lOf. 

23.  From  5|  subtract  3^. 

24.  From  7  subtract  dj%.  3/g, 

25.  A  grocer  bought  123^  lb.  of  butter.  After  selling 
52 1  lb.  of  it,  how  much  had  he  left  ?  70§  lb. 

26.  A  farmer  cut  24f  T.  of  hay  from  two  meadows,  cutting 
9||  T.  from  one  of  them.  How  much  hay  did  he  cut  from 
the  other?  HUT. 

27.  From  19f  subtract  f§.  l^jh 

28.  If  I  have  5|^  acres  of  land,  and  I  sell  f  of  an  acre,  how 
much  land  have  I  left  ?  J^,^-^  acres. 

29.  From  a  bin  which  contained  4|  bu.  of  potatoes,  a  house- 
keeper used  2|  bu.    How  many  potatoes  were  left  in  the  bin  ? 


MULTIPLICATION.  177 

30.  A  merchant  buys  boots  at  $5}  a  pair,  and  sells  them  at 
$7  a  pail'.     What  are  his  profits  on  each  pair  ? 

31.  A  bin  that  will  hold  190  bu.,  contains  104|  bu.  of  wheat. 
How  many  bushels  more  will  the  bin  hold  ?  SS^. 

32.  From  171  subtract  ^. 

33.  From  103  subtract  40j\.  62j%: 

34.  Wishing  to  pay  my  butcher  $8|,  I  hand  him  a  lO-doUar 
bill.    How  much  change  ought  I  to  receive  ?  ^1^, 


SECTION  V. 

CASE     I. 
The  Multiplicand  a  Fraction. 

225.  Ex.  Multiply  |  by  4. 
Explanation. — Since  a  fraction  is  multi-      boltttion. 
plied  by  multiplying    its  numerator   (See      2x4=| 
208, 1),  we  multiply  2,  the  numerator  of  the 
given  fraction,  by  4,  and  under  the  product  write  the 
denominator. 

mOBJLEMS. 

1.  Multiply  2%  by  5.  ^. 

2.  What  is  the  product  of  /^  multiplied  by  7  ? 

3.  How  much  is  3  times  f  ?  ^^  or  f. 

4.  If  a  man  and  team  can  plow  j\  of  an  acre  in  an  hour, 
how  much  land  can  they  plow  in  4  hours  ?  ^  A. 

5.  How  much  is  7  times  f  ?  ^^-,  or  2f. 

6.  How  much  will  8  bushels  of  oats  cost,  at  $|  a  bushel  ? 

7.  How  much  will  5  cloth  caps  cost,  at  $|  a  piece  ? 

8.  How  far  will  a  locomotive  run  in  34  minutes,  at  the  rate 
of  I  of  a  mile  a  minute  ?  15  mi. 

9.  If  Y^o  of  an  acre  will  pasture  1  cow  through  the  summer, 
how  many  acres  will  pasture  18  cows  ?  16^  A, 


178  FRACTIONS. 

10.  A  carpenter  built  15  lengths  of  board  fence,  and  each 
length  was  f  |  of  a  rod  long.    How  long  was  the  fence  ? 

11.  Multiply  f  by  11,  ii  by  9,  and  j\  by  15. 

12.  How  much  is  6  times  ff  lb.  ?    8  times  j\  doz.  ? 

4^^  lb. ;  4-3  do^- 

C^SE     II. 
The  Multiplier  a  Fraction. 

226.  Ex.  1.  Multiply  15  by  f  ;  that  is,  find  §  of  15. 
Explanation.  —  |  is  equal  to  2 

times  I,  and  |   is  the    result  of  solution. 

dividing  1  by  3.    Hence,  to  get  |  of  15  -^  3  =  5 

15,  we  first  divide  it  by  3  to  find  j  5  x  2  =  10 

of  it,  and  then  multiply  the  result,  _  • 

5,  by  2,  to  find  f  of  it.  S^^^^'  ^^  ^  f =^^- 

Ex.  2.  Multiply  15  by  j%  ;  or,' find  j%  of  15. 
Explanation. — ^We  first  di-  solution.        ^ 

vide  15  by  12  to  find  J^  of         15  -^  12  =  H 
15,    and   then    multiply    the     15  y^^—T_5  —  q_3  _.gi 
result,  i|,  by  5,  to  find  {^  of     '^  '^         '^       ' 

15.  Hence,  15  x  /^  =  6j^. 

227.  Hence,  to  multiply  any  number  by  a  fraction,  we 
Divide  the  multiplicand  by  the  denominator j  and  multi- 
ply the  result  hy  the  numerator. 

PMOJBZJSMS. 

13.  Multiply  18  by  I ;  that  is,  find  f  of  18.  IS. 

14.  What  is  the  product  of  45  multiplied  by  §  ? 

15.  How  much  is  ^  of  43  yards  of  ribbon  ?  24-  yd. 

16.  Multiply  7  by  j\.  j-f,  or  Sj\. 

17.  I  bought  300  lb.  of  nails,  and  used  \  of  them  in  building 
a  bam.    How  many  nails  did  I  use  ?  262^  lb. 

18.  A  fat  ox  weighed  1,173  lb.,  and,  when  killed,  the  beef 
weighed  \\  as  much.    How  much  did  the  beef  weigh  ? 


MULTIPLICATION.  179 

19.  Last  year  I  gathered  13  bushels  of  plums  from  my  gar- 
den, and  f  of  them  were  damsons.  How  many  damson 
plums  had  I  ? 

20.  Multiply  57  by  j'V,  and  23  by  i§.  26§,  2^. 

21.  A  and  B  bought  a  mowing  machine  for  $145,  A  paying 
i^g  of  the  cost,  and  B  {'^.     How  much  did  each  man  pay  ? 

C-A.se    III. 

Both  Factors  Fractions. 

228.  Ex.  Multiply  §  by  f  ;  or,  find  |  of  |. 

Explanation. — ^We  first  divide  |  by  3  to  bolutiok. 

find^  of  |.     This  we  do  by  multiplying         4      _  _4^ 

the  denominator  by  3.  (See  208,  n.)    We        4  x2  _  _8^ 

then  multiply  the  result,  /_,  by  2,  to  find  |       ^^      ~  '^ 

of  |.  Hence,  ixf,or§off  =  -,%. 

229.  Hence,  to  multiply  a  fraction  by  a  fraction,  we 
Multiply  the  numerators  together  for  a  new  numerator, 
and  the  denominators  together  for  a  new  denominator. 

The  word  of  between  fractions  signifies  multiplica- 
tion ;  thus,  I  of  J  =  I  X  |. 

PROBLEMS. 

22.  Multiply  I  by  f.  ^. 

23.  WhaHs  the  product  of  |  x  |  ?  ^, 

24.  Multiply  ^  by  I ;  ^,  by  ^. 

25.  How  much  will  f  gal.  of  syrup  cost,  at  $f  a  gal.  ?   $^. 

26.  How  much  will  |  yd.  of  brown  linen  cost,  at  $|  a  yd.  ? 

27.  How  much  is  f  of  \\  lb.  ?  |^  ^. 

28.  How  much  is  |  of  ^  of  an  apple  ?  §■  of  an  apple. 

29.  How  much  wood  is  f  of  |  of  a  cord  ? 

30.  What  part  of  a  melon  is  f  of  f  of  a  melon  ? 

31.  Three  men  own  a  factory  in  equal  shares.  How  much 
of  the  factory  does  each  man  own  ?  If  one  man  sells  \  of  his 
share,  what  part  of  the  factory  does  he  sell  ?  i-. 


180  FRACTIONS. 

33.  A  man  who  owned  {  of  a  sMp  sold  f  of  his  share.  How 
much  of  the  ship  did  he  sell  ?  -fj^, 

33.  I  of  y'^o  =  what  fraction  ? 

34.  What  is  the  product  of  f  x  f  x  f  ? 

We  multiply  all  the  numerators  together  for  the 
numerator  of  the  product,  and  all  the  denominators 
together  for  the  denominator  of  the  product. 

35.  What  is  the  product  of  |  x  f  x  y3_  ?         ^^o_^  ^  ^^. 

36.  ixfx4xf  =  how  many  ?  /-j-. 

37.  ^  of  f  is  what  part  of  1  ?  /y. 

38.  I  of  i  of  J  is  what  part  of  1  ?  /g-. 

39.  What  is  the  product  of  |,  ^,  and  ^^  ? 

0-A.SE3    I"Vr. 
One  or  both  Factors  Mixed  Numbers. 

230.  Ex.  Multiply  3^  by  2|. 
Explanation. — ^We  first  reduce  solxttion. 

the  mixed  numbers  to  improper       3}  =  ?  and  2?  =  § 
fractions,  and  then  multiply,  as     7x-8  =  56_92__.9i 

in  Casein.  ^     ^     2     J     n! 

Hence,  S^x  2§=^  9^. 

PBOBZEMS. 

40.  What  is  the  product  of  6  times  3f  ?  22 j-. 

41.  How  much  will  9  barrels  of  flour  cost,  at  $10|  a  barrel  ? 
43.  If  a  man  builds  4|  rd.  of  stone  fence  in  1  day,  how  much 

can  he  build  in  13  days  ?  60§  rd. 

43.  Multiply  8  by  4|.  37^. 

44.  How  much  will  3|  lb.  of  opium  cost,  at  $9  a  lb.  ? 

45.  How  much  will  |  yd.  of  vesting  cost,  at  $3|  a  yd.  ? 

46.  Multiply  3,--^^  by  2f.  8§§. 

47.  What  is  the  product  of  4|  x  3/y  ?  17^. 

48.  Multiply  9f  by  ^^ ;  4^  by  11|.  5|| ;  W- 

49.  How  many  sq.  rd.  in  a  field  36|  rd.  long,  and  21  f  rd. 
wide?  791  fi. 


DIVISION.  181 


SECTION  VI. 
^irisioj\r. 

CASE     I. 
The  Divisor  an  Integer. 
231.  Ex.  Divide  f  by  4. 
Explanation.  —  To  divide 
I  by  4,  is  to  find  |  of  |.    To      ^  ^      joLtrxioN^  _  e  -  2 
do  this,  we  write  \  of  f ,  and      I  "^  ^  —  I  o^  I  =  3%  =  I 
multiply,   as  in   Case  HI.,  tt  o       /  _  o 

MultipHcation.    (See  230.)  ■^^^^®'  ^  -^  ■♦  -  #• 

232*  Hence,  when  the  divisor  is  an  integer,  we 
Write  it  as  the  denominator  of  a  fraction  with  1  for  a 

numerator t  and  multiply  the  given  fraction  by  the  fraction 

thusformed. 

FROBZJEMS. 

1.  Divide  f  by  4.  /^,  or  ^. 

2.  Divide  j\  by  5.    Divide  |f  by  16.  /j.     ^. 

3.  If  4  lb.  of  sugar  cost  $||,  how  much  does  1  lb.  cost? 

4.  Six  boys  gathered  ||  of  a  bushel  of  chestnuts,  and  shared 
them  equally.    How  many  chestnuts  in  1  boy's  share  ? 

5.  A  butcher  packed  f  of  a  ton  of  pork  in  8  barrels.    How 
much  did  he  put  in  each  barrel  ?  ^^  T. 

6.  A  seamstress  used  |  of  a  yard  of  linen  in  making  9  col- 
lars.    How  much  linen  did  she  use  for  each  collar  ? 

7.  If  4  oz.  of  iudigo  cost  $f ,  what  is  the  price  of  1  oz.  ?  $^y, 

8.  Divide  2f  by  8. 

Before  dividing,  reduce  the  mixed  number  to  an  im- 
proper fraction. 

9.  What  is  the  quotient  of  3|  divided  by  4  ?  ^. 

10.  If  a  teamster  draws  3|  cords  of  stone  at  15  loads,  how 
much  does  he  draw  at  each  load  ?  ^  cd. 


182 


FRACTIONS. 


11.  What  is  the  quotient  of  15|  divided  by  18  ?        |-||. 

12.  If  13  boxes  of  strawberries  cost  $3^,  how  much  does  1 
box  cost  ? 

13.  If  7  men  can  bind  22}f  acres  of  wheat  in  one  day,  how 
much  can  1  man  bind  ?  S^  A. 

14.  How  many  times  is  9  contained  in  414  ^  -4#f  • 

15.  Divide  400^^  by  23.  11  j\. 


C^SE    II. 
The  Divisor  a  Fraction. 


5-^| 


Ex.  1.  Divide  5  by  | 

Explanation.  —  Since  the 
quotient  is  not  changed  by 
multiplying  both  dividend 
and  divisor  by  the  same  num- 
ber (See  208,  HI),  we  multi- 
ply them  both  by  3,  and  thus 
obtain  15  for  a  new  divi- 
dend, and  2  for  a  new  divi- 
sor. Then,  15  -^  2  =  ^-  = 
T^,  the  required  quotient. 

Ex.  2.  Divide  |  by  |. 

Explanation. — We 
first  multiply  both 
dividend  and  divisor 
by  5,  the  denomina- 
tor of  the  divisor, 
(See  148),  and  then 
divide  the  new  divi- 
dend, J^^-,  by  the  new 
divisor,  2,  as  in  Case  I. 

(See  Manual,  page  220.) 


FIRST   SOLTTTION. 

5  X  3  =  15 

I  X  3=    2 

15  --  2  =  -y  = 

Hence,  5  ~  % 


=  7i. 


SECOND    SOLTTTIOII. 


=  15-^2 

Hence,  5  • 


f  =  n- 


Jj4-^2 


FIE8T    SOLUTION. 

I  X  5  =  -'/- 
|x5=  2 

—    1    ^f   1  5_  _ 
_  5  Ol  -4-  — 

Hence,  |  -^ 


SECOND    SOLUTION. 


Hence,  f  -=-  #  = 


1? 


=1? 


DIVISION.  183 

234.  From  these  examples  it  will  be  seen  that,  to 
divide  by  a  fraction,  we 

Multiply  the  dividend  by  the  denominator  of  the  divisor, 
and  divide  the  result  by  the  numerator. 

FMOBIjEMS 

16.  Divide  3  by  f .  7|. 

17.  Divide  7  by  ^,  and  6  by  f.  21;  8, 

18.  At  $1  a  bushel,  how  many  bushels  of  apples  can  be 
bought  for  $5  ?  ^I  hi. 

19.  The  dividend  is  9,  and  the  divisor  is  f.  What  is  the 
quotient  ?  Jfi^. 

20.  What  is  the  quotient  of  f^  divided  by  f  ?  1^. 

21.  How  many  times  is  f  contained  in  ff  ? 

22.  How  much  ribbon,  at  %^^  a  yard,  can  be  bought  for  $f  ? 

23.  K  a  horse  walks  a  mile  in  y*j  of  an  hour,  how  far  will  he 
walk  in  8  hours  ?  30  miles. 

24.  At  $1  a  pound,  how  much  candy  can  be  bought  for  $|  ? 

25.  Divide  3}  by  4|. 

Before  dividing,  reduce  the  mixed  numbers  to  im- 
proper fractions.     Thus,  Si-J-  4|  =  -L«-  -t-  ^-  =  4f  =  U- 

26.  What  is  the  quotient  of  8  -j-  2|  ?  2^, 

27.  Divide  \l  by  3|.  /^. 

28.  The  dividend  is  5f,  and  the  divisor  is  f.  What  is  the 
quotient?  i^|.. 

29.  A  shoe-dealer  paid  $150  for  a  case  of  boots,  at  $6|  a 
pair.     How  many  pairs  of  boots  were  in  the  case  I 

30.  A  man  whose  daily  wages  were  $2  j,  received  at  the  end 
of  the  week  $12f .     How  many  days  had  he  worked  ?       5f . 

31.  I  paid  %ll  for  5i  lb.  of  rice.   What  was  the  price  per  lb.  ? 

32.  How  many  rolls  of  wall-paper,  each  containing  4^ 
sq.  yd.,  will  be  required  to  cover  58^  sq.  yd.  of  wall  ?       13, 

33.  What  is  the  quotient  of  f  divided  by  f  ? 

34.  Divide  1  by  i| ;  h^  by  ^  3.5.  .  ^_5_. 

35.  What  is  the  quotient  of  15|  -r-  24^  ?  144. 


184:  FRACTIONS. 

SECTION  VII. 

CAJ\rCBZ  ZiATIOJSr. 

ca.se    I. 

In  Multiplication. 

235.  What  is  the  product  of  f ,  4,  and  \\  ? 

Explanation. — ^In  the 
First  Solution  we  re-      ^     ^     2TL'''''T''l  10 
duce  the  product,  ^^^^,      5X7X22-  tVs^  -  suf  =  W 
to  its  lowest  terms,  by  bbcond  boltttion. 

dividing  both  dividend  13 

and  divisor  by  4,  and  ^  x  s^  x  ^f  =  J| 

both  terms  of  the  result 

thus  obtained  by  7.  But,  in  the  Second  Solution,  we  di- 
vide the  numerator,  4,  and  the  denominator,  8,  by  4,  and 
the  numerator,  21,  and  the  denominator,  7,  by  7.  We 
then  multiply  the  remaining  numbers  in  the  numer- 
ators together  for  the  numerator  of  the  product,  and 
the  remaining  numbers  in  the  denominators  together 
for  the  denominator  of  the  product.  The  results  in  the 
two  solutions  are  the  same.  (See  Manual,  page  220.) 

236.  The  process  of  dividing  a  numerator  and  a 
denominator  by  any  number,  either  in  the  same  frac- 
tion, or  in  fractions  which  are  to  be  multiplied  together, 
is  Cancellation. 

In  multiJ)Ucation  of  fractions,  whenever  a  numerator 
and  a  denominator  contain  the  same  factor,  the  process 
can  be  shortened  by  Cancellation.  In  this  case,  the 
product  will  always  be  in  its  lowest  terms. 

PJROBIjEMS. 

1.  Multiply  4  by  I ;  I  by  /o ;  and  f  by  |f.      /^,|,  and  J^. 
3.  What  is  the  product  of  f ,  |,  and  f  ?  ^. 

3.  Multiply  I,  I,  §,  and  |  together. 


Division.  185 

4.  How  much  is  |  of  |  of  |  of  i\  ? 

5.  f  X  2|  X  1|  X  I  =  what  fraction?  f. 

6.  j\  of  I  of  I  of  f  is  what  part  of  1  ?  ^, 

7.  How  much  must  I  pay  for  Q^\  ft.  of  gas-pipe,  at  $f  a  ft.  ? 

8.  How  many  sq.  ft.  of  oil-cloth  will  be  required  to  cover  an 
office  table  3f  ft.  long,  and  2|  ft.  wide  ?  10. 

9.  How  much  will  If^  yd.  of  linen  cost,  at  $if  a  yd.  ? 

10.  How  much  must  I  pay  for  cutting  f  of  ^  cd.  of  wood,  at 
|of$i|acord?  $f 

C^SE     II. 

In  Division. 

237.  Ex.  1.  Divide  |  by  |. 

Explanation.  —  In 
this  Solution  we  have  ^'^^'^  s^^^™^- 

multiplied    the    nu-     ^  •  ^  — '^'   •   ^  —  s  oi  -^  —  2^ 
merator  of  the  divi- 
dend by  the  denominator  of  the  divisor  (5x4  =  20), 
and  the  denominator  of  the  dividend  by  the  numerator 
of  the  divisor  (9  x  3  =  27). 

If  we  change  the  places  of  the  second  solution. 

terms  of  the  divisor,  and  multi-     f-^4=f^3=i? 
ply  the  dividend  by  |,  the  frac- 
tion thus  formed,  we  shall  multiply  the  same  numbers 
together  as  in  the  First  Solution.     This  is  shown  in 
the  Second  Solution.     Hence, 

238.  In  dividing  by  a  fraction,  we  may 
Change  the  places  of  the  terms  of  the  divisor ,  and  mul- 
tiply the  dividend  by  the  fraction  thus  formed. 
Ex.  2.  Divide  f^  by  j|. 
Explanation. — ^We  first  in-  solution. 

vert    the     divisor  —  that     is,     _5  _i.  is  _  J      \a  _  4 
change  the  places  of  the  terms     ^  ^  *    *  ^  ~  ^J^    ^~ 


-and  then  proceed  as  in  Case  I. 
P 


186  FRACTIONS. 

PMOBIjEMS. 

11.  Divide  f  by  |,  and  f  by  f .  /^,  and  f . 

12.  How  many  times  is  ^^  contained  m^^^'i  2^-. 

13.  I  paid  $58^  for  building  3|  rods  of  door-yard  fence.  How 
much  did  it  cost  me  a  rod  ?  $15^. 

14.  Divide  11  by  4^? 

15.  How  many  brooms  can  be  made  from  22i  lb.  of  broom- 
corn,  if  y5^  lb.  is  used  for  1  broom  ?  6  doz. 

16.  A  hardware  dealer  paid  $45  for  sheep  shears,  at  %^-^  a 
pair.     How  many  pairs  did  he  buy  ? 

17.  How  many  boxes,  each  5|  in,  long,  41  in.  wide,  and  3  in. 
deep,  will  be  required  to  hold  as  much  as  one  box  8|  in.  long, 
8^  in.  wide,  and  6  in.  deep  ?  6. 

18.  Divide  i?_  of  f  by  i  of  I  of  %.  /^. 


SECTION  VIII. 

^HO'SZBMS  IJ\r  FHACTIOJVS, 

1.  A  provision  dealer  sold  7  barrels  of  pork  for  $113|. 
What  was  the  price  per  barrel  ?  $16{, 

2.  How  many  cubic  feet  are  there  in  a  block  of  marble  4^ 
ft.  long.  If  ft.  wide,  and  \  ft.  thick?  ^if. 

3.  How  much  will  1  pound  of  sugar  cost,  if  4|  pounds  cost 

$H?  $2%' 

4.  An  errand  boy  earned  $f  on  Monday,  %\  on  Tuesday,  and 
II  on  Wednesday.    How  much  did  he  earn  in  the  three  days  ? 

5.  Reduce  |f  to  its  lowest  terms. 

6.  In  17y^,  how  many  elevenths  ? 

7.  How  much  can  a  man  earn  in  f  of  a  day,  at  $1|  a  day  ? 

8.  A  grocer  buys  cheese  at  %^§-^  a  pound,  and  sells  it  at  %\ 
a  pound.     How  much  does  he  gain  per  pound  ? 

9.  If  you  pronounce  63  words  each  minute  in  reading  aloud, 
how  long  vdll  it  take  you  to  read  a  chapter  containuig  5,796 
words  ?  111.82  min. 


REVIEW    PROBLEMS.  187 

10.  The  sum  of  two  numbers  is  83§,  and  one  of  them  is  47^. 
What  is  the  other  ?  36^^^. 

11.  The  difference  of  two  numbers  is  2f,  and  the  greater 
number  is  61  ^q.    What  is  the  less  number  ?  ^^/^. 

13.  The  difference  of  two  numbers  is  j-^^,  and  the  less  num- 
ber is  ^\.     What  is  the  greater  number  ?  i-||-. 

13.  How  many  lights  of  glass,  each  containing  |  of  a  square 
foot,  are  there  in  a  box  of  glass  which  contains  50  square  feet  ? 

14.  Reduce  3/y  to  forty-fourths. 

15.  Reduce  f ,  f ,  and  f  to  similar  fractions. 

16.  Reduce  ^%\^-2  to  a  mixed  number  ? 

17.  If  llf  doz.  eggs  cost  %'^j%,  how  much  will  13f  doz.  cost  ? 

18.  How  many  pieces  of  stone  flagging,  each  23  ft.  square, 
will  it  take  to  make  a  sidewalk  148^  ft.  long  and  5^  fb.  wide  ? 

19.  What  part  of  19^  is  18yL  ?  i.^. 
.20.  One  Saturday  morning,  Frank  caught  five  fish,  the  first 

of  which  weighed  3|\  lb.,  the  second  3|  lb.,  the  third  2|  lb., 
and  the  fourth  and  fifth  each  2|  lb.  What  was  the  weight 
of  all? 

21.  A  farmer  sold  f  of  his  land,  and  afterward  bought  37f 
acres.  He  then  had  1122^^  acres.  How  much  land  had  he  at 
first?  ^  12JfA. 

22.  What  is  the  difference  between  f^  of  18f  and  |  of  17|  ? 

23.  How  much  will  a  turkey  weighing  8}^  pounds  cost,  at 
$/;japound?  ^%VV 

24.  If  you  sleep  1-^^  hours  each  day,  how  much  time  do  you 
spend  in  sleep  in  1  year  ?  16  wh  IQ-ji^  h. 

25.  A  grocer  bought  a  barrel  of  sugar,  and  after  selling  y\ 
of  it,  ^  of  it,  and  -^%  of  it,  he  had  119  pounds  left.  How  many 
pounds  were  in  the  barrel  at  first  ?  272. 

26.  Reduce  ^Vq  ^^  its  lowest  terms. 

27.  A  laborer  laid  189  ,V  rods  of  tile-drain  in  13|  days. 
How  much  did  he  lay  each  day  ?  ISf  rods- 

28.  I  paid  12  cents  a  pound  for  llf  pounds  of  beef,  but  -^ 
of  it  was  bone.  As  the  bone  was  worth  nothing,  how  much  a 
pound  did  the  meat  cost  me  ?  16§^  cents. 


188  FRACTIONS. 

29.  If  a  man  can  earn  $17.50  in  26  days,  when  he  works  10 
hours  a  day,  how  much  can  he  earn  in  19  days,  when  he  works 
12  hours  a  day  ?  $62.70. 

30.  How  much  will  375  pounds  of  bone-dust  cost,  at  $38  a 
ton?  $7,125. 

31.  William  is  ^-^  as  old  as  his  father,  his  father  is  ^  as  old 
as  his  grandfather,  and  his  grandfather  is  72  years  old.  How 
old  is  William  ?  12  years. 

32.  Two  men  dug  a  well  in  five  days,  digging  11|  ft.  the 
first  day,  6|  ft.  the  second,  5i  ft.  the  third,  3|  ft.  the  fourth, 
and  3/2  ft.  the  fifth.     How  deep  was  the  well  ? 

33.  If  Andrew  can  run  968f  rods  in  25  minutes,  and  Rich- 
ard can  run  963  rods  in  24  minutes,  which  can  run  the  faster  ? 

34.  If  they  start  together,  and  run  in  the  same  direction  for 
18^  minutes,  how  far  will  they  be  apart  ?  ^-^/g-  '"'o^^- 

35.  The  minuend  is  201  f,  and  the  subtrahend  \^.  Wha*  is 
the  remainder  ? 

36.  The  dividend  is  {^  of  If,  and  the  divisor  is  |  of  87|. 
What  is  the  quotient  ?  j^j. 

37.  The  sum  of  three  numbers  is  357  j^^,  and  two  of  them 
are  265 1  and  \\.     What  is  the  third  number?  90^^. 

38.  A  sewing  girl  earns  $|  a  day,  and  pays  $2|  a  week  for 
her  board.  How  much  money  will  she  have  at  the  end  of  13 
weeks,  after  paying  for  her  board  ?  $16^. 

39.  How  many  square  feet  are  there  in  a  board  16  ft.  long 
and  I  ft.  wide  ? 

40.  Four  boys  spent  one  Saturday  in  gathering  walnuts. 
On  dividing  them,  Albert  received  \  of  all  they  gathered, 
Eobert  |-\,  Thomas  f^^,  and  David  17 1  quarts.  How  many 
walnuts  did  each  of  the  other  boys  have  ? 

41.  How  many  walnuts  did  they  gather  in  all  ? 

21m.  1  pk.  S  qt. 

42.  If  12|  acres  of  wheat  can  be  cut  with  a  reaper  in  1  day, 
how  many  acres  can  be  cut  in  3,4  days  ?  Jj^js' 


CHAPTER  Y. 
PERCENTAGE. 

SECTION  I. 

239«  In  business  transactions,  hundredths  of  any  thing 
or  number  are  commonly  called  Fer  Cent.  Thus,  .04  is 
4  per  ceni.,  .09  is  9  per  cent.,  .48  is  48  per  cent.   Hence, 

I.  Any  per  cent,  of  a  thing  or  number  is  so  many  hun- 
dredths of  that  thing  or  number  ;  and, 

II.  Per  cent,  may  always  be  written  as  a  decimal. 

240.  This  character,  ^,  placed  after  a  number,  is  the 
Commercial  Sign  of  per  cent.  Thus,  15%  signifies  15 
hundredths,  and  is  read  15  per  cent. 

EXER  CIS  ES. 

1.  Write,  decimally,  8  per  cent,,  3  per  cent.,  5  per  cent. 

2.  Write  14  per  cent,  17  per  cent,  28  per  cent,  33  per  cent., 
40  per  cent.,  65  per  cent.,  10  per  cent 

3.  Read  the  following  decimals  as  per  cent. :  .07,  .19,  .30 
.42,  .50,  .69,  .06,  .99,  .75. 

4.  Eirst  read,  and  then  write,  decimally,  12^,  29^,  63^,  90^. 

5.  Read,  and  then  write,  1^  9^,  10^,  56^,  47^. 

6.  Write,  with  the  sign  of  per  cent.,  the  decimals  in  Exer- 
cise 3. 

241.  125^  is  written,  decimally,  1.25,  308^  is  written 
3.08. 

VALUES.  OF   FRACTIONS   IN   DECIMALS. 


^  =  .5,    and  ^^  =  .005 
I  =  .25,  and  Ifo  =  .0025 
f  =  .75,  and  |^  =  .0075 
I  —.  .2,    and  4^  =  .002 


I  =  .125,  and  Ifo  =  .00125 
f  =  .375,  and  Ifo  =  .00375 
f  =  .625,  and  |^  =  .00625 

I  =  .875,  and  l^  =  .00875 


190  PERCENTAGE. 

12^^  is  written  .12^  or  .125,  i%  or  .25%  is  written 
.001  or  .0025.     Hence, 

I.  To  express  per  cent,  decimolly,  two  decimal  figures 
are  always  required. 

II.  To  express  more  than  99  per  cent.y  an  integer  or 
a  mixed  decimal  number  is  required. 

III.  To  express  parts  of  1  per  cent.,  decimal  figures  or 
fractions  at  the  right  of  hundredths  are  required. 

JEXEMCISES. 

7.  Write,  decimally,  114^,  159^,  237^,  475^  108^,  200^. 

8.  Read  .038,  .0425,  .165,  .43f,  .311,  .05^. 

9.  Write  6,%  16]^,  10^^,  18.7^,  22.5^,  31.25^  {^,  Ifc. 


SECTION   II. 

GBJSTB'RAZ   dTTZIC^ATIOJSrS. 

242»  Per  cent,  may  be  applied  to  any  number,  great 
or  small,  concrete  or  abstract. 

243.  The  process  of  finding  any  per  cent,  of  a  num- 
ber is  Percentage.  The  result  of  the  computation  is 
also  called  Percentage. 

244.  The  number  expressing  the  per  cent,  is  the 
Bate,  or  Bale  Per  Gent. 

245.  The  number  on  which  the  percentage  is  com- 
puted is  the  Base. 

Ex.  How  much  is  15^  of  125  ?  '°''''"''''' 
Explanation.— Since  15^  of  a  ^"f^  ^'^ 

number  is  .15  of  that  number,  -^ 

we  multiply  125  by  .15,  as  in  -1^25 

MultipHcation  of  Decimals.  -^^^^ 

lo.  f  O  Percentage. 
246.  ^ule  for  'Percentaffe, 

Multiply  the  base  by  the  rate  expressed  decimally. 


GENERAL     APPLICATIONS.  191 

PBOBI^EMS* 

1.  Of  a  flock  of  125  sheep,  4^  were  killed  by  dogs.  How 
many  sheep  were  killed  ?  5. 

3.  At  a  school  of  125  pupils,  the  average  daily  attendance 
is  88^  of  the  whole  number.    What  is  the  daily  attendance  ? 

3.  From  a  cask  of  44  gallons  of  oil,  5^  leaked  out.  How 
much  oil  leaked  out  ?  2.2  gallom. 

4.  Crystal  Lake  covers  87^  of  a  square  mile.  How  many 
acres  does  it  cover  ?  556.8. 

5.  Of  a  cargo  of  14,865  bushels  of  wheat,  12^  was  injured 
by  water.    How  many  bushels  were  injured  ?  1783.8. 

6.  A  piece  of  cotton  cloth  containing  42  yards,  shrank  6^ 
in  bleaching.     How  many  yards  did  it  then  contain  ? 

100^  of  a  number  is  the  whole  of  it.  100^  —  6^  = 
94^;  and  94^  of  42  yd.  =  S9.4S  yd. 

7.  A  fruit-grower  set  out  250  peach-trees,  but  18^  per  cent. 
of  them  died.     How  many  trees  lived  ?  205. 

8.  A  steamboat  company  bought  5,280  cords  of  wood,  of 
which  21^  was  hickory,  33^  was  beach,  and  the  balance  was 
maple.     How  many  cords  of  each  kind  did  they  buy  ? 

9.  A  man  husked  284  bushels  of  com,  and-  received  12^^ 
of  it  for  his  labor.    How  much  com  did  he  receive  ? 


SECTION   III. 

COMMISSIOJV. 


247.  A  person  who  buys  and  sells  goods  for  another, 
receiving  for  his  services  a  certain  <fc  on  the  value  of 
the  goods  bought  or  sold,  is  an  Agent  or  Gommissicm- 
Merchant 

248.  The  sum  paid  the  agent  or  commission-mer- 
chant for  his  services  is  Commission. 


192  PERCENTAGE. 

Ex.  A  real-estate  agent  sold  a  house  and  lot  for 
$2,275.     How  much  did  he  receive 
for  making  the  sale,  at  2^  com-         solution. 
mission  ?  $2275  Base. 

02 
Explanation. — Since  he  received        '- — 

a  commission  of  2%,  he  received  .02         $45.50  commission. 
of  $2,275,  which  is  $45.50. 

PJROBJOEMS. 

1.  An  auctioneer  sold  farm  produce  and  stock  to  the 
amount  of  $1,185,  and  charged  5^  commission.  How  much 
did  he  receive  ?  $59.25. 

2.  A  commission-merchant  sold  500  barrels  of  flour  for 
$4,750.     How  much  was  his  commission,  at  3^  ?  $95. 

8.  A  land  agent  paid  out  $2,970  for  western  land,  on  a 
commission  of  5^.  How  much  did  his  commission  amount 
to  ?  $U8.50. 

4.  A  canvassing  agent  received  40^  for  selling  pictures. 
His  receipts  were  $106  in  Jan.,  $95  in  Feb.,  and  $126  in  Mar. 
How  much  did  he  earn  each  month  ? 

5.  My  agent  purchases  for  me  2,680  bu.  of  wheat,  at  $1,125 
a  bushel,  on  a  commission  of  2^^.  How  much  is  his  com- 
mission? "  $75.37^. 

6.  I  sold  45,000  pounds  of  cotton,  at  $.17  a  pound,  on  a 
commission  of  f  ^.  How  much  did  my  commission  amoimt 
to?  $28.68s 


SECTION   IV. 

JJVS  77  ^  A  J\r  C  B, 

249.  A  company  that,  for  a  certain  sum,  agrees  to 
pay  for  property,  if  it  is  destroyed  by  fire  or  lost  at 
sea,  is  an  Insurance  Company;  and 

250.  The  agreement  is  an  Insurance. 

251.  The  sum  of  money  paid  to  secure  an  insurance 
is  the  Premium. 


INSURANCE.  198 

252.  The  premium  is  usually  computed  at  a  certain 
per  cent,  on  the  sum  for  wliich  the  property  is  insured. 

JPMOBJ^JEMS. 

1.  How  much  will  it  cost  to  insure  a   solution  op  pkoblem  i. 
dwelling  for  $850,  at  2fo  ?  1850 

2.  What   premium    must  a  carpenter  ~ — 

pay  to  get  his  shop  insured  for  $975,  at       $17.00  Premium. 
3^?  $29.25. 

3.  A  hotel  and  furniture  are  insured  for  $45,000,  at  2^. 
What  is  the  annual  or  yearly  premium  ?  $900. 

4.  A  cargo  of  flour,  shipped  from  New  York  to  Liverpool, 
is  insured  for  $9,000,  at  ^^.     How  much  is  the  premium  ? 

5.  How  much  will  it  cost  a  year  to  insure  a  school-house  for 
$1,350,  at  an  annual  premium  of  f^? 

6.  How  much  will  it  cost  for  3  years  ?  $28.12^. 

7.  An  agent  insured  a  flouring-mill  for  $6,750,   at  1{^. 
What  premium  did  he  receive  ?  $8J/,.37^-. 

8.  A  ship  was  insured  for  a  voyage  from  Boston  to  San 
Francisco  for  $32,500,  at  2^^.     What  was  the  premium  ? 


SECTION  V. 

TllOI^IT    M.JV^     ZOSS. 

253.  When  anything  is  sold  for  more  than  it  cost, 
the  gain  or  increase  in  its  value  is  Frofit ;  and 

254.  When  it  is  sold  for  less  than  it  cost,  the  decrease 
in  its  value  is  jLoss. 

255.  Profit  and  Loss  are  commonly  estimated  at 
some  per  cent,  upon  the  cost  price  of  the  property. 

Ex.  A  merchant  bought  broadcloth  at  $2.50  a  yard, 
and  sold  it  so  as  to  gain  40^.  How  much  did  he  gain 
on  a  yard  ?    At  what  price  per  yard  did  he  sell  it  ? 


SOLUTION. 


194  PERCENTAGE 

Explanation. — ^At  40^,  the  gain 
on  $2.50  is  .40  of  $2.50,  which  is 

$1.00  ;  and  $2.50,  the  cost  price,  ^^'^^  ^°'*  ^^ 

+  $1.00,  the    gain,  =  $3.50,    the    — 

selling  price.  $1-00    Gain, 

Had  the  cloth  been  sold  at  a  loss  ^'^^ 

of  40^,  the  $1.00  would  be  loss,  $3.50    SeiUng  Price. 
and  the    selling  price  would    be 

found  by  subtracting  the  loss  from  the  cost. 

Pit  OBZJEMS. 

1.  A  mechanic  built  a  house  at  a  cost  of  $1,500,  and  in  sell- 
ing it,  gained  12^.     How  much  did  he  gain  ?  $180. 

2.  A  grocer  sells  sugar  that  costs  him  $.12  a  pound,  at  25^ 
profit.    How  much  does  he  make  on  1  pound  ?  $.03. 

3.  In  selling  kerosene  that  costs  $.48  a  gallon,  at  56|^^ 
profit,  how  much  is  the  gain  on  1  gallon  ?  $.27. 

4.  I  bought  a  horse  for  $180,  and  sold  him  at  a  loss  of  12|^. 
How  much  did  I  lose  by  the  bargain  ?  $22.50. 

5.  A  farmer  paid  $6,000  for  75  acres  of  land,  and  sold  it  at 
7^^  less  than  he  gave  for  it.     How  much  was  his  loss  ? 

6.  A  merchant  sells  delaines  which  cost  $.31}  a  yard,  at  a 
loss  of  20^.     How  much  does  he  lose  on  a  yard  ?         $.06±. 

7.  At  what  price  must  I  sell  a  sewing-machine  that  cost  me 
$45,  to  gain  25^  ?  $56.25. 

8.  A  carpet  dealer  bought  carpeting  at  $1.25  a  yard,  and 
marked  it  to  be  sold  at  60^  above  cost.  At  what  price  did  he 
mark  it  ?  $2.00. 

9.  If  coal  that  costs  $6.75  a  ton  is  sold  at  2i%  below  cost, 
what  is  the  selling  price  ?  $5.13. 

10.  What  price  must  I  put  on  a  bureau  that  cost  me  $15 
to  manufacture,  in  order  to  gain  40^  ? 

11.  A  man  paid  $300  for  a  village  lot,  $1,600  for  building  a 
house  upon  it,  and  $200  for  other  improvements.  He  then 
offered  to  sell  the  property  at  10^  below  cost  ?  What  was  hia 
asking  price  ?  $1,890. 


INTEREST.  195 

SECTION  VI. 

IJV*T£J^BS  T. 

I.  If  a  person  hires  a  house  or  a  fann,,lie  pays 
sometliing  for  the  use  of  it. 

So,  also,  if  a  person  borrows  money,  when  he  pays 
the  debt,  he  pays  the  sum  borrowed  and  an  additional 
sum  for  the  use  of  it ;  and 

If  a  person  pays  a  debt  after  it  is  due,  he  pays  not 
only  the  amount  of  the  debt,  but  an  additional  sum  for 
the  use  of  that  amount. 

257.  The  sum  paid  for  the  use  of  money  is  Interest. 

258.  The  sum  for  the  use  of  which  the  interest  is 
paid  is  the  Principal. 

259.  The  sum  of  the  principal  and  interest  is  the 
Amount. 

260.  The  rate,  or  the  interest  of  1  dollar  for  1  year,  is 
always  some  per  cent,  of  the  princij)al. 

In  most  States,  the  rate  per  cent,  is  established  by 

law. 

oa.se    I. 

Interest  for  Years. 

261.  Ex.  1.  What  is  the  interest  of  $395  for  1  year, 

Sit  b^  .  B0LT7TI0N. 

Explanation. — At  Q%,  the  interest  $395  Principal. 

is  .06  of  the  principal,  and  .06  of  1395  .06  Rate. 

is  $23.70.  $23.70  interest. 

Ex.  2.  What  is   the    interest  of  solution. 

$135.25  for  3  years,  at  Q%  ?  $135.25 

Explanation. — The  interest  for  3  .06 

years  is  3  times  as  much  as  the  in-  $8.1150  int.  for  i  yr. 

terest  for  1  year,  or  3  times  $8,115,  3 

which  is  $24,345.  $24,345    int.  for  s  yr. 


196  PERCENTAGE. 

Ex.   3.    What  is  the   amount  of  solutioit. 

$350  for  5  years,  at  1%  ?  $^^^ 

Explanation.  —  The    interest    of  ^oTHo 

$350  for  5  years,  at  1%,  is  $122.50  ;  ^^^'^g 

and  $122.50  (the  interest),  +  $350  -^too^h  t  .     . 

n^  •       '     \\  &,Annrr\         jlv.  $>122.5U    Interest 

(the    principal),    =    $472.50,     the         35Q       p^,„,jp,, 

^^OU-Di'  1472:50  Amount. 

JPJ2  OBZJEMS. 

1.  What  is  the  interest  of  $25  for  1  year,  at  6^  ?       $1.50. 

2.  Find  the  interest  of  $132  for  1  year,  at  5fo.  $6.60. 

3.  Find  the  interest  of  $76.50  for  1  year,  at  7^.       $5,355. 

4.  What  is  the  interest  of  $216.25  for  3  years,  at  8fo  ?  $51.90. 

5.  A  man  paid  a  debt  of  $188.65,  with  interest  at  10^,  1 
year  after  it  was  due.     What  amount  did  he  pay  ?    $207,515. 

6.  What  is  the  amount  of  $3,750  for  3  years,  at  5^  ? 

7.  February  11,  1864,  I  borrowed  $250.     How  much  did 
the  debt  amount  to,  February  11,  1867,  interest  at  6^? 

8.  What  is  the  interest  of  $560.10  for  2  years,  at  6i^  ? 

C^SE     II. 
Interest  for  Months. 

262.  Ex.  What  is  the  interest  of  $84.18  for  7  months, 

at  6^? 

Explanation. — The  interest  for  soltttion. 
1  year  is  $5.0508  ;  and  the  inter-  $84.18 
est  for  7  months,  or  ^^  of  a  year,  '^^ 

is  y\  of  $5.0508,  which  is  $2.9463.        $5.0508  mt.  for  1  yr. 

To  find  ^2  of  $5.0508,  the  in-      I 

terest  for  1  year,  we  multiply  it      $35.3556  [  12 

by  the  numerator,  7,  and  divide       $2.9463  intfor^vyr. 

the  product  by  the  denominator, 

12  (see  227) ;  but  7  is  the  given  number  of  months, 

and  12  is  the  number  of  months  in  a  year.     Hence, 


INTEEEST.  197 

To  find  the  interest  for  any  number  of  months. 
Multiply  the  interest  for  1  year  by  the  given  number  of 
months,  and  divide  the  product  by  12. 

(See  Manual,  page  220.) 

The  answers  to  the  remaining  problems  in  this  Chap- 
ter are  given  in  accordance  with  the  principle  stated  in 
Art.  159. 

PHOBZEMS. 

9.  What  is  the  interest  of  $153.17  for  3  months,  at  6^  ? 

10.  What  is  the  interest  of  $18.72  for  5  months,  at  4^  ? 

11.  Find  the  mterest  of  $584.34  for  1  yr.  4  mo.,  at  7^. 

(1  yr.  4  mo.  =  16  mo.)  $54.53. 

13.  If  I  have  $1876.50  at  interest  for  3  yr.  5  mo.,  at  6^,  how 
much  interest  will  I  receive  ?  $272.09. 

13.  What  is  the  amomit  of  $394.25  for  6  mo.,  at  5^  ? 

14.  K  I  give  my  note  for  $275,  Jan.  11,  1867,  and  pay  it 
Feb.  11,  1868,  with  7^  interest,  how  much  do  I  pay  ? 

15.  If  I  borrow  $782,  at  12^  interest,  and  pay  the  debt  in 
1  month,  how  much  do  I  pay  ?  $739.32. 

CA.SE     III. 
Interest  for  Days. 

263.  In  computing  interest,  30  days  are  called  a 
month.     Hence, 

Every  3  days  are  j^^,  or  .1  of  a  month,  and 
Every  1  day  is  \  of  j^^,  or  .0|  of  a  month. 

264.  Ex.  1.  What  is  the  interest  of  $675     bolittion. 
for  18  days,  at  7^?  ^g^g* 

^  Explanation. — Since  every  3  days  are  .1  .07 

of  a  month,  18  days  are  .6  of  a  month.  $47^ 

We  therefore  multiply  $47.25,  the  interest  .6 

for  1  year,  by  .6,  and  divide  the  product  $28  850  1  J^^S^ 

by  12,  as  in  Case  II.  $2i3625~^    "^^ 

03r         ^  « 


198  PERCENTAG 


SOLUTION. 

$169.44 
.06 


Ex.   2.    What  is  tlie    interest    of 
$169.44  for  1  yr.  3  mo.  17  da.,  at  6^? 

Explanation. — Since  1   yr,    3   mo.  |10  1664 

are  15  mo.,  and  17  da.  are  .5§  of  a  15.5^ 

mo.,   the   whole  time  is   15.5f   mo.  SS888~ 

We  therefore  multiply  $10.1664,  the  38888 

interest  for  1  year,  by  15. 5 1,  and  di-  508320 

vide  the  product  by  12,  as  in  Case  508320 

n.,  and  the  result,  $13.19,  is  the  in-  101664 

terest  required.     In  multiplying  by  $158.25696  [12 

I,  we  take  |  of  the  multiplicand  twice.  $13,188 

TearSy  months,  and  days  can  be  expressed  as  months  and 
tenths  of  a  month, 

16.  What  is  the  interest  of  $116.25  for  24  days,  at  6^?  $.JiB5. 

17.  If  I  borrow  $819  for  20  days,  at  8^,  how  much  interest 
must  I  pay  ?  $3.64"  . 

18.  How  much  interest  will  I  have  to  pay,  at  7^,  on  a  loan 
of  $1296,  for  9  mo.  15  da.  ?  $1!1.82. 

19.  What  is  the  interest  of  $936  for  3  yr.  2  mo.  29  da.,  at 
10^  ?  $sos,H- 

20.  Find  the  interest  of  $718  for  1  yr.  14  da.,  at  6^. 

21.  What  is  the  interest  of  $48,  from  Nov.  23,  1866,  to 
Dec.  8,  1867,  at  7^  ?  ^'^•^^• 


General  ^ule  for  Interest. 

I.  For  1  year,  Multiply  the  principal  by  the  rate. 
n.  For  2  or  more  years.  Multiply  the  interest  for  1 
year  by  the  number  of  years. 

III.  For  any  other  time.  Multiply  the  interest  for  1 
year  by  the  time  expressed  in  months  and  tenths  of  a 
month,  and  divide  the  product  by  12. 

IV.  For  the  amount.  Add  the  interest  to  the  principal. 


INTEREST.  199 

PHOBJLEMS. 

27.  How  much  interest  must  I  pay  for  the  use  of  $756.50 
for  5  years,  at  7^  ?  $261^.775. 

28.  A  note  of  $1834.75,  dated  Oct.  9,  1867,  was  paid  Oct.  9, 
1868,  with  interest  at  6^.     What  was  the  amount  paid  ? 

29.  There  is  a  mortgage  on  my  house  and  lot  for  $1244, 
with  interest  at  7^.     How  much  interest  is  due  annually  ? 

80.  A  teacher  in  St.  Louis  bought  a  house  and  lot  for  $3750, 
and  paid  for  it  at  the  end  of  3  years,  with  interest  at  6^. 
What  amount  did  he  pay  ?  ^^4^5. 

31.  What  is  the  interest  of  $752.50  for  8  months,  at  %  ? 

32.  What  is  the  interest  of  $87.36,  at  10^,  from  Feb.  10, 
1866,  to  Oct.  10,  1867  ?  $U.56. 

33.  A  young  man  bought  a  watch  for  $85,  and  paid  for  it 
in  9  months,  with  interest  at  Ifc.     How  much  did  he  pay  ? 


SECTION   VII. 

1.  On  opening  a  box  of  128  panes  of  glass,  a  glazier  found 
^\<fo  of  them  broken.     How  many  panes  were  broken  ?        8. 

2.  I  had  $1,756  in  accounts,  and  I  have  collected  78^  of 
them.    What  sum  have  I  collected  ?  $1369.68. 

3.  A  sewing-machine  agent  in  one  year  sold  285  machines, 
at  $60  each.  How  much  did  his  commission  amount  to,  at 
25^?  $Jf,275. 

4.  A  produce  dealer  sold  1,756  pounds  of  butter,  at  $.28  a 
pound,  for  a  country  merchant.  How  much  was  his  commis- 
sion, at  5^  ?  ■  $2JfMJt. 

5.  At  1^^,  what  premium  must  I  pay  for  an  insurance  of 
$1,500  on  my  dwelling?  $22.50. 

6.  At  f^,  what  premium  must  a  farmer  pay  annually  for 
an  insurance  of  $800  on  his  house,  $900  on  his  bams,  and 
$1,100  on  his  grains  and  hay? 


200  PERCENTAGE. 

7.  How  much  premium  will  he  pay  in  5  years  ?         $105. 

8.  A  country  store  is  insured  for  $1,350,  and  the  goods  are 
insured  for  $3,250,  at  1|^.  How  much  is  the  annual  premium 
on  the  store  and  goods  ?  $78.12^. 

9.  A  farmer  bought  a  reaper  for  $140,  and  gave  his  note  in 
payment  for  it.  He  paid  the  note  4  years  afterward,  with  in- 
terest at  10^.     How  much  did  he  pay  ?  $196. 

10.  A  note  for  $417.12,  dated  Aug.  23,  1867,  was  due  Sept. 
23,  1868,  with  interest  at  8^.  What  sum  was  required  to  take 
up  the  note  ?  $453.27. 

11.  What  is  the  amount  of  $10843.40  for  1  yr.  7  mo.,  at 
10^  ?  $12560.27. 

12.  Find  the  interest  of  $162  for  4  mo.  9  da.,  at  6|^.  $3.77. 

13.  What  is  the  interest  on  $3,000  for  1  yr.  5  mo.  2  da.,  at 
7.3^?  $311.^7. 

14.  What  is  the  amount  of  $792  for  17  da.,  at  5^  ?  $793.87. 

15.  Dec.  15, 1864,  a  farmer  mortgaged  his  farm  for  $4,850, 
and  Sept.  15,  1867,  he  paid  the  mortgage,  with  6^  interest. 
What  amount  did  he  pay  ?  $5650.25. 

16.  In  a  school  of  140  pupils  45^  of  them  are  boys,  and 
55^  are  girls.     How  many  boys  belong  to  the  school  ? 

17.  How  many  girls  are  in  the  school  ?  77. 

18.  A  young  man  invested  $3,450  in  business,  and  lost  17^ 
of  it  the  first  year.  How  much  money  had  he  left  ill  the 
business  ?  $2863.50. 

19.  A  produce  dealer  bought  wheat  at  $1.60  per  bushel, 
and  sold  it  at  a  loss  of  10^^.     At  what  price  did  he  sell  it  ? 

20.  If  I  invest  $2,500  in  bank  stock,  and  sell  it  at  an  advance 
of  Qfc,  for  what  sum  do  I  sell  ? 

21.  April  11,  1867,  a  man  bought  a  canal  boat  for  $1,140, 
and  Nov.  11,  1868,  he  paid  for  it,  with  interest  at  7^.  How 
much  did  he  pay  ?  $1266.35. 


MISCELLANEOUS     PEOBLEMS.  201 


CHAPTER  YI. 

MISC£;LZ;)iJV:EOUS   1>^0SZBMS; 

Mmhracing   all   the  Principles   and   MetJwds   of  Computation 
in  the  Preceding  Cliapters,    (See  Manual,  page  220.) 

1.  A  drover  paid  $35  each  for  32  head  of  cattle,  and  $4'} 
each  for  23  head.  How  many  cattle  did  he  buy,  and  how 
much  did  they  cost  him  ?  They  cost  him  $2,086. 

2.  A  farmer  sold  a  farm  of  96.23  acres  at  $85  an  acre,  and 
afterward  bought  another  of  123.47  acres  for  $52.50  an  acre. 
Which  fann  came  to  the  more  money  ?   How  much  the  more  ? 

The  first  farm;  $1697.37^. 

3.  A  dealer  in  garden  seeds  put  up  8  bu.  2  pk.  5  qt.  of 
peas  in  papers  holding  1  pt.  each.  How  many  papers  did  he 
have  ?  654" 

4.  How  many  years,  months,  and  days  old  are  you  to-day  ? 

5.  How  long  a  time  has  elapsed  since  the  Declaration  ol 
American  Independence,  which  was  made  July  4, 1776  ? 

6.  How  many  bushels  of  lime  can  be  bought  for  $7.35,  at 
$.20  a  bushel  ?  36.75. 

7.  A  lady  paid  $52.50  for  28  yards  of  carpetmg.  What 
was  the  price  per  yard  ? 

8.  At  what  price  per  lb.  must  I  sell  84:5  lb.  of  raisins,  to 
receive  $37.18  for  them  ?  $.44. 

9.  If  1  lb.  of  cheese  can  be  made  from  3f  qt.  of  milk,  hov/ 
many  lb.  can  be  made  from  35 1  qt.  ?  9|. 

10.  A  gardener  has  12  hives  of  bees,  and  last  summer  he 
obtained  S\  lb.  of  honey  from  each  hive.  How  much  honey 
did  he  get  in  all  ? 

11.  A  hop  grower  bought  3,875  hop-poles,  at  $22  a  thou- 
sand.   How  much  did  be  pay  for  them  ? 

Since  1,000  hop-poles  cost  $22,  1  hop-pole  must  cost 
y^^oo  (or  .001)  of  $22,  which  is  $.022  ;  and  3,875  hop- 
poles,  at  $.022  apiece,  must  cost  3,875  times  $.022, 
which  is  ^85.25. 

Q 


202  MISCELLANEOUS     PROBLEMS. 

13.  A  nursery-man  sold  144  apple-trees,  at  $12  a  hundrecl. 
How  much  did  he  receive  for  them  ?  $17.28. 

13.  In  building  a  house,  I  used  21,375  feet  of  clapboards, 
for  which  I  paid  $35  per  thousand.  How  much  did  they  cost 
me?  $7Jj.8.12^. 

14.  What  will  be  the  cost  of  7  pieces  of  sheeting,  each  con- 
taining 39  yards,  at  $.37^  a  yard  ?  $102.37^, 

15.  How  many  cubic  feet  in  a  plank  16  ft.  long,  1  ft.  wide, 
and  .25  ft.  thick  ? 

16.  A  roadway  500  ft.  long,  and  16  ft.  wide,  is  to  be  cov- 
ered 1.5  ft.  deep  with  gravel.  How  many  cu.  yd.  of  gravel 
will  be  required  ?  J^J^Jf.  cu.  yd.  Jf,  cu.ft. 

17.  Two  men  start  from  the  same  place  at  the  same  time, 
and  travel,  one  43.82  miles,  and  the  other  34.57  miles  a  day. 
How  far  apart  will  they  be  at  the  end  of  7.5  days,  if  they 
travel  in  opposite  directions  ?  587.925  miles. 

18.  How  far  apart  will  they  be  in  the  same  time,  if  they 
travel  in  the  same  direction  ?  69.375  miles. 

19.  How  many  pieces  of  wall-paper,  each  9  yd.  long  and  \ 
yd.  wide,  will  be  required  to  paper  81  sq.  yd.  of  wall  ?     18. 

20.  A  regiment  was  mustered  into  the  army  with  938  men. 
During  the  term  of  service,  93  men  were  killed,  76  died  of 
sickness,  214  were  mustered  out,  295  were  taken  prisoners, 
183  deserted,  and  349  new  recruits  joined  the  regiment.  How 
many  men  belonged  to  the  regiment  when  its  term  of  service 
expired  ?  J,,26. 

21.  If  1.25  lb.  of  rags  are  required  for  1  yd.  of  rag  carpet, 
how  many  yards  of  carpet  can  be  made  from  30  lb.  of  rags  ? 

22.  How  many  miles  of  telegraph  line  can  be  constructed 
with  6,116  poles,  if  22  poles  are  set  to  the  mile  ? 

23.  At  the  beginning  of  the  year,  the  population  of  a  cer- 
tain city  was  31,675.  During  the  year  the  number  of  deaths 
was  764,  and  the  number  of  births  803 ;  1,236  people  removed 
from  the  city,  and  1,394  removed  to  it.  What  was  the  popu- 
lation at  the  end  of  the  year  ?  31, 872. 


MISCELLANEOUS    PROBLEMS.  203 

24.  A  freight  train  of  13  cars  is  loaded  with  96  barrels  of 
flour  to  each  car.  How  many  barrels  of  flour  on  the  train  ? 
How  many  tons  ?  122.301^  tons. 

25.  An  ice  dealer  delivered  to  his  customers  1,296  lb.  of  ice 
daily  for  27  days,  1,794  lb.  daily  for  26  days,  2,146  lb.  daily 
for  56  days,  1,834  lb.  daily  for  24  days,  and  1,310  lb.  daily  for 
21  days.     How  many  lbs.  of  ice  did  he  deliver  ? 

26.  How  many  tons  of  ice  did  he  deliver  ?  136.669. 

27.  How  many  pounds  was  his  average  daily  delivery  ? 

28.  In  13,427  pints  of  beans,  are  how  many  bushels  ? 

29.  Into  how  many  building  lots  can  12  acres  of  land  be 
divided,  each  lot  being  4  rods  front  and  10  rods  deep  ? 

30.  How  many  bricks  can  be  put  into  a  pile  12  ft.  long, 
6  ft.  wide,  and  4  ft.  high,  each  bnck  being  8  in.  long,  4  in. 

wide,  and  2  in.  thick  ?      (See  Manual,  page  220.)  7,  776. 

31.  Reduce  204,080  cu.  in.  to  higher  denominations. 

^  cu.  yd.  10  cu.ft.  176  cu.  in. 

32.  Reduce  11  mi.  4  yd.  2  ft.  to  feet.  68, 09 J^  ft. 

33.  If  a  load  of  hay  with  the  wagon  weighs  3,165  pounds, 
and  the  wagon  alone  weighs  1,249  pounds,  how  much  is  the 
hay  worth,  at  ^11  a  ton  ?  $10,538. 

34.  A  grocer  paid  $60.75  for  oranges,  at  $6.75  a  box.  How 
many  boxes  did  he  buy  ? 

35.  A  paper  manufacturer  paid  $231.40  for  rags,  at  $65  a 
ton.     How  many  rags  did  he  buy  ?  3.56  tons. 

36.  A  barrel  inspector  examined  400  barrels,  and  condemned 
^\fo  of  them.     How  many  barrels  bore  inspection  ?         390. 

37.  A  provision  dealer  bought  pork  at  $12.50  a  hundred, 
and  sold  it  at  an  advance  of  20^.  At  what  price  did  he  sell 
it  ?  $15  a  hundred. 

38.  If  I  sell  railroad  stock  which  cost  me  $2,500,  at  a  loss 
of  8|^,  how  much  do  I  receive  for  it  ?  $2287.50. 

39.  How  much  will  it  cost  to  insure  a  factory  for  $28,000, 
at  2\%  premium  ? 


204  MISCELLANEOUS    PROBLEMS. 

40.  The  owners  of  the  brig,  Ivanhoe,  paid  1|^  for  an 
insurance  of  $17,750  for  a  single  voyage  to  the  "West  Indies. 
How  much  did  the  insurance  cost  them  ?  $199.68^. 

41.  How  much  must  be  paid  for  an  insurance  of  $8,650  on 
a  cargo  of  wheat  from  Milwaukee  to  Oswego,  at  |^  ?    $54-06 j^. 

42.  A  dairyman  sent  4,320  lb.  of  butter  to  a  commission- 
merchant,  whose  rate  of  commission  was  4^  on  his  sales.  He 
sold  the  butter  for  $.37^  a  pound.  How  much  did  the  dairy- 
man realize  ?  $1555.20. 

43.  An  agent  receives  40^  commission  for  selling  maps. 
How  much  will  his  commission  be  on  sales  amounting  to 
$3,280,  and  how  much  will  the  map  publisher  receive  ? 

$1,312;  $1,968. 

44.  What  is  the  interest  of  $341.08  for  3  yr.  10  mo.,  at 
5i^  ?  $71.91. 

45.  To  how  much  will  $74.18  amount  in  2  yr.  2  mo.,  at 
dfc  ?  88.64^. 

46.  Find  the  amount  of  $250  for  5  yr.  7  mo.,  at  6^  ? 

47.  The  owners  of  a  vessel  that  was  overdue  from  Liver- 
pool, fearing  that  she  was  lost,  paid  18^  for  an  insurance  of 
$32,000  upon  her.     How  much  did  the  insurance  cost  them  ? 

48.  A  gentleman  whose  house  cost  him  $18,000,  had  it 
insured  for  $14,000,  at  |^  premium.  Should  the  house  bum 
down,  what  would  be  his  entire  loss  ?  $4122.50. 

49.  How  many  cubic  feet  in  a  pile  of  20  planks,  each  12  ft. 
long,  10  in.  wide,  and  2  in.  thick  ?  33^. 

50.  A  farmer  raised  8.5  acres  of  flax,  which  yielded  850 
pounds  to  the  acre.  How  much  was  the  crop  worth,  at  $.06 
a  pound  ?  $433.50. 

51.  How  many  acres  are  there  in  100  miles  of  a  road  4  rods 
wide  ?  800. 

52.  In  a  box  containing  50  sq.  ft.  of  window-glass  how 
maAy  panes  are  there,  each  pane  being  10  in.  long  and  8  in. 
wide  ?  90. 

53.  If  5  cows  eat  2y^^  T.  of  hay  in  5  wk.,  how  much  hay  will 
1  cow  eat  in  1  wk.  ?  175  lb. 


MISCELLANEOUS    PROBLEMS.  205 

54.  How  many  yards  of  carpeting  will  it  take  to  cover  the 
floor  of  a  parlor  6|  yd.  long  and  5^-  yd.  wide?  35§r, 

55.  Keduce  V/  ^^  a  mixed  number. 

56.  A  wood-chopper  cut  three  trees  into  cordwood.  The 
first  tree  made  3|  cd.,  the  second  4j%,  and  the  third  5^  cd. 
How  much  wood  did  the  three  trees  make  ? 

57.  What  will  be  the  cost  of  850  handbills,  at  the  rate  of 
$3.50  for  the  first  hundred,  and  $1.25  for  each  succeeding  hun- 
dred? $12.87^, 

58.  Bought  18,280  ft.  of  pine  flooring,  at  $24  a  thousand. 
How  much  did  it  cost  ?  $J,S8.72. 

59.  A  bushel  contains  2150.42  cubic  inches.  How  many 
cubic  inches  in  87.5  bushels  ?  188161.75. 

60.  How  many  bushels  in  135476.46  cubic  inches  ? 

61.  A  farmer  has  a  bin  7  ft.  long,  6  ft.  wide,  and  4  ft.  deep. 
How  many  bushels  will  it  hold  ?  135  hi.,  nearly. 

62.  A  gallon  liquid  measure  contains  231  cubic  inches. 
How  many  cubic  inches  in  275  gallons  ? 

63.  How  many  gallons  in  13051.5  cubic  inches  ?         56.5. 

64.  A  box  11  in.  long,  7  in.  wide,  and  3  in.  deep,  will  hold 
how  many  gallons  ? 

65.  What  is  the  capacity  in  gallons  of  a  cistern  8  ft.  long,  8 
ft.  wide,  and  7  ft.  deep  ?  3351j\. 

66.  A  tanner  has  a  vat  that  will  hold  1075.21  gallons. 
How  many  bushels  will  it  hold  ?  117.5. 

67.  At  2.^^  commission,  how  much  will  a  commission-mer- 
chant receive  for  selling  750  barrels  of  pork,  at  $21  a  barrel  ? 

68.  A  traveling  agent  sold  1,600  young  apple-trees,  at  $16 
a  hundred.  How  much  did  his  commission  amoimt  to,  at 
25^?  $64. 

69.  How  much  will  $290  amount  to  in  3  years,  at  8^  in- 
terest ? 

70.  What  is  the  interest  of  $2,750  for  8  mo.,  at  7.3^  ? 

71.  At  7^,  how  much  interest  must  I  pay  for  the  use  of 
$195.75,  from  May  13  to  November  8  following  ?         $6.66. 


206  MISCELLANEOUS     PKOBLEMS. 

72.  A  seamstress  bought  a  sewing-machine  for  $75,  paying 
$i5  down,  and  the  balance  in  4  months,  with  interest  at  7^. 
What  was  the  amount  of  the  last  payment  ?  $61.^0. 

73.  A  note  for  $417.13,  dated  Jan.  10,  1866,  was  paid  Dec. 
14,  1867,  with  Qfo  interest.     What  was  the  amount  paid  ? 

74.  A  music  dealer  sold  a  piano,  which  cost  him  $324,  for 
33^^  above  cost.     How  much  did  he  get  for  it  ?  $4S2. 

75.  If  23  lb.  of  starch  can  be  made  from  1  bu.  of  corn,  how 
much  corn  will  be  required  for  63,365  lb.  of  starch  ? 

76.  A  railroad  company  having  343  A.  of  woodland,  cut  from 
28.5  A.  58.75  cd.  of  wood  per  acre,  from  93.3  A.  52.5  cd.  per 
acre,  and  from  112.7  A.  48.25  cd.  per  acre.  How  much  wood 
was  cut,  and  how  many  acres  of  wood  remained  standing  ? 

120104  cd.;  108.5  A. 

77.  A  farmer  sowed  13^  bu.  of  wheat  on  10^  acres  of  land, 
and  the  yield  was  14^  bu.  per  acre.  How  much  wheat  did  he 
get  above  his  seed  ?  139^  iu. 

78.  How  many  tons  will  500  barrels  of  flour  weigh  ?     49. 

79.  23  teams  were  employed  47  days  in  drawing  earth  for 
a  railroad  embankment,  each  team  averaging  15  cu.  yd.  24  cu. 
ft.  a  day.     How  much  earth  was  in  the  embankment  ? 

17175  cu.  yd.  ^  cu.ft. 

80.  How  much  can  I  save  in  a  year,  if  I  earn  $100  a  month 
for  10  months,  and  spend  $68.63  every  month  ?        $176.^. 

81.  How  many  feet  in  a  board  16  ft.  long  and  9  in.  wide  ? 

82.  A  manufacturer  finds  that  merino  wool  loses  .42  of  its 
weight  in  cleaning.  How  much  weight  will  2345.5  pounds 
lose  ?  985.11  'pounds. 

83.  Find  the  interest  of  $25  for  7  yr.  3  mo.  24  da.,  at  10^. 

-■$18.29. 

84.  What  is  the  amount  of  $110.62  for  3  yr.  7  mo.  28  da., 
at  7^?  $138.97. 

85.  What  sum  must  be  paid  to  cancel  a  debt  of  $219.16, 
which  has  been  due  1  yr.  6  mo.  14  da.,  at  the  rate  of  interest 
in  this  State  ? 

86.  I  bought  a  house  for  $1,850,  and  afterward  sold  it  at  an 
advance  of  30^.  How  much  was  my  gain,  and  for  how  much 
did  I  sell  it  ?  Qain^  $555  ;  Belling  price^  $^,405. 


MISCELLANEOUS    PROBLEMS.  207 

87.  The  sum  of  three  parts  is  100  mi.,  and  two  of  the  parts 
are  33  mi.  225  rcL  2  yd.  2  ft.,  and  17  mi.  90  rd.  3  yd.  What 
is  the  third  part  ? 

88.  131  +  I  +  5^  +  f  4-  what  number  =  4rt ,%  ?       27^h 

89.  After  traveling  21^  and  18^^  of  a  journey  of  425  miles, 
what  i  of  the  journey  had  I  yet  to  travel  ?  How  many  miles 
had  1  yet  to  travel  ?  ^^9.25  miles. 

90.  How  much  will  |^  of  a  ton  of  hay  cost,  at  $7.50  a  ton  ? 

91.  Brass  is  composed  of  copper  and  zinc.  In  a  quantity 
of  brass  that  weighed  f  ^  of  a  ton,  the  copper  weighed  ^\  of  a 
ton.     What  was  the  weight  of  the  zinc  ?  ^^  T. 

92.  If  a  family  burn  f|  of  a  cord  of  wood  in  30  days,  how 
much  wood  will  they  burn  in  1  day?  tIs  ^* 

93.  From  Tvhat  number  must  I  subtract  984,006  to  obtain  a 
remainder  of  9,276,985  ?  10,260,991. 

94.  The  subtrahend  is  six  thousand  and  twenty-four  ten- 
thoqsanths,  and  the  remainder  four  thousand  ninety-six  hun- 
dred-thousandths.    What  is  the  minuend  ? 

95.  The  minuend  is  17  cu.  yd.,  and  the  remainder  16  cu.  yd. 
1,596  cu.  in.     What  is  the  subtrahend  ?      26  cu.ft.  132  cu.  in. 

96.  If  16  shoemakers  make  5^0  pairs  of  shoes  in  16.25  days, 
how  many  pairs  can  1  workman  make  in  1  day  ?  2. 

97.  A  farmer  exchanged  85  lb.  of  butter  at  $.21  a  pound, 
for  flannel  at  $.35  a  yard.     How  many  yards  did  he  receive  ? 

98.  If  a  young  man  smokes  1,284  cigars  in  a  year,  how 
much  will  his  year's  supply  cost  him,  at  $28.50  a  thousand  ? 

$36.59. 

99.  What  will  be  the  cost  of  17,890  feet  of  hemlock  lumber, 
at  $11  a  thousand  ? 

100.  What  number,  multiplied  by  7,296,  will  produce 
292,518,528  ? 

101.  The  multiplicand  is  19^,  and  the  product  319?.  What 
is  the  multiplier  ?  16^, 

102.  The  product  of  three  factors  is  29.2923,  and  two  of  the 
factoi-s  are  11.4  and  285.5.     What  is  the  third  factor  ? 

103.  How  many  square  inches  in  a  mirror  5  ft.  3  in.  high  and 
26  in.  wide  ? 


208  MISCELLANEOUS    PROBLEMS. 

104.  In  a  village  lot  66  ft.  front  by  133  ft.  deep,  are  how 
many  sq.  rd.  ?  32. 

105.  K  16  men  in  17.25  days  can  mine  529.92  tons  of  iron 
ore,  how  miicli  ore  can  1  man  mine  in  1  day  ?  1.92  T. 

106.  A  clergyman  had  his  household  furniture  insured  for 
$850,  and  his  library  for  $650,  at  |^.  What  premium  did  he 
pay  annually  ?  $11.25. 

107.  The  dividend  is  9,  and  the  quotient  576.  What  is  the 
divisor  ?  .015625. 

108.  What  number  divided  by  19^ "3  will  give  a  quotient  of 

••■1  21   * 

109.  A  farmer  cures  his  hams  by  the  following  recipe  :  For 
every  100  lb.  of  meat,  9  lb.  of  salt,  5  oz.  of  saltpeter,  4  oz.  of 
ground  pepper,  and  1  qt.  of  molasses.  What  quantity  of  each 
ingredient  must  he  use  for  675  pounds  of  meat  ? 

Salt^  60.75  lb. ;  saltpeter,  83.75  oz. ;  ground  pepper,  27  os. ; 
molasses^  6.75  qt. 

110.  In  salting  beef,  the  same  farmer  uses  the  following 
recipe:  For  every  100  lb.  of  meat,  6  qt.  of  salt,  1  qt.  of 
molasses,  and  4  oz.  of  saltpeter.  What  quantity  of  each  in- 
gredient must  he  use  for  380  lb.  of  meat  ?  ^ 

Salt,  22.8  qt. ;  molasses,  3.8  qt. ;  saltpeter,  15.2  oz. 

111.  Which  is  the  more  advantageous,  to  borrow  $175  at 
7^  interest  to  pay  house-rent  m  advance,  or  to  pay  $200  rent 
at  the  end  of  the  year  ?    How  much  the  more  advantageous  ? 

To  lorrow,  hy  $12.75. 

112.  A  man  can  hire  a  farm  of  97  acres  for  $500  per  annum, 
or  he  can  buy  it  for  $70  an  acre.  If  money  is  worth  6^,  which 
is  the  cheaper  course,  and  how  much  the  cheaper  ? 

To  luy  the  farm,  Iry  $92.60  per  annum. 

113.  A  farmer  bought  80  sheep,  at  $4.20  a  head,  giving  his 
note,  payable  in  6  mo.,  at  7^.  At  the  end  of  the  6  mo.  he 
sold  the  sheep  at  $5.25  a  head  cash,  and  paid  the  note.  How 
much  did  he  get  for  the  keeping  of  the  sheep  ?  $72.24. 

114.  A  man  had  his  life  insured  for  $5,000  at  the  age  of  29 
years,  and  died  at  the  age  of  47.  His  yearly  premiums  aver- 
aged $71.85.  How  much  more  did  his  family  receive  than  he 
had  paid  ? 


MANUAL 


OP 


METHODS   AND  SUGGESTIONS. 


A  Word  with  Teachers. — This  Manual  is  intended  to  give  you 
hints  and  suggestions  rather  than  detailed  methods  of  instruction ; 
and  to  call  your  attention  to  those  points  which  require  your  special 
efforts,  if  you  would  secure  that  thoroughness  in  your  pupils  which 
is  essential  to  real  progress.  It  is  not  intended  to  lay  down  pre- 
scribed forms  for  you  to  follow,  hut  to  give  you  such  hints  and  sug- 
gestions as  will  enable  you  to  work  out  details  of  methods  of  in- 
struction, and  to  adapt  them  to  your  classes,  in  accordance  with 
your  methods  of  thought. 

Use  of  Objects. — Children  gain  ideas  more  readily  by  perception 
than  by  reflection.  You  should,  therefore,  illustrate  the  subjects  of 
lessons  by  Visible  Objects  whenever  this  is  practicable.    For  example  : 

In  Notation. — A  child  may  be  aided  in  comprehending  the  idea  of 
a  ten  and  of  a  hundred  by  the  use  of  counters.  Take  a  quantity  of 
beans,  or  other  suitable  objects,  make  a  pile  of  ten  of  them  for  a  ten, 
and  ten  such  piles  near  together  for  ten  tens  or  a  hundred. 

In  Addition.— Illustrate  the  addition  of  two  numbers,  as  37  and  48, 
thus :  3  piles  of  tens  and  7  counters  or  ones  may  represent  37 ;  and  4 
piles  of  tens  and  8  counters  or  ones  may  represent  48.  Then  put- 
ting the  7  counters  and  8  counters  together,  there  will  be  enough 
counters  for  1  ten  and  5  counters  or  ones  more :  The  1  ten,  3  tens, 
and  4  tens,  together,  are  8  tens,  and  the  8  tens  and  5  ones,  are  85. 

In  Subtraction. — The  process  may  be  illustrated  in  a  similar 
manner. 

In  Compound  Numbers. — Let  the  pupils  see  and  handle  the  various 
measures,  and  they  will  get  clear  ideas  of  a  quart,  a  bushel,  a  foot,  a 
pound,  or  any  other  denomination.  Let  them  measure  water,  and 
see  that  2  pints  are  1  quart,  and  4  quarts  1  gallon.  Let  them  exercise 
their  judgment  upon  the  capacity  of  vessels,  by  estimating  how  much 
a  pail,  a  pan,  a  pitcher,  a  cup,  or  a  bottle,  will  hold ;  and  then  require 
them  to  measure  the  vessel  to  correct  their  judgment.  In  the  same 
manner  let  them  measure  sand,  or  com,  to  become  familiar  with  the 
denominations  of  dry  measure.  Also,  let  them  by  trial  see  if  a  quart 
liquid  measure  is  the  same  as  a  quart  dry  measure,  etc. 

To  make  them  familiar  with  the  denominations  of  distance,  let 
them  draw  lines,  an  inch,  a  foot,  and  a  yard  long,  upon  the  black- 
hoard  :  judge  of  the  width,  length,  and  height  of  the  room,  or  the 


210  MANUAL    FOK    TEACHERS. 

house ;  and  then  measure  the  various  distances  to  correct  their  judg- 
ment. Again,  for  greater  distances,  let  them  measure  the  distance  of 
a  rod  along  the  fence,  then  40  rods  or  1  eighth  of  a  mile  on  the  road. 
Let  them  estimate  the  distances  between  different  objects  in  the  room, 
about  the  yard,  and  along  the  road,  and  afterward  measure  them. 

If  no  other  measures  can  be  obtained,  cut  from  a  lath,  or  any  other 
straight  stick,  a  piece  1  inch  long ;  another  1  foot  long,  and  mark  it 
off  into  inches  with  a  knife  or  pencil ;  and  another  1  yard  long,  and 
mark  it  off  into  feet.  A  cord  or  a  piece  of  rope  may  be  used  for  the 
measure  of  a  rod. 

In  square  measure,  have  a  square  inch,  a  square  foot,  and  a  square 
yard.  You  can  show  that  these  might  be  used  to  measure  with,  but 
that  it  is  more  convenient  to  take  dimensions  with  linear  measures. 

The  measure  of  a  cubic  inch  may  be  made  by  cutting  out  a  piece  of 
pasteboard  in  this  form,  cutting  about  one 
half  through  the  thickness  where  the  lines 
cross  it,  folding  it  together,  and  fastening 
the  joining  edges.  Larger  cubes  may  be 
made  in  the  same  manner.  A  box  1  foot 
long,  1  foot  wide,  and  1  foot  high,  is  a 
measure  of  a  cubic  foot.  The  surface 
should  be  marked  off  into  square  inches. 

If  your  schoolroom  is  not  supplied  with  scales  and  weights,  you 
can  prepare  a  few  weights  by  making  packages  or  bags  of  shot,  sand, 
or  pebbles,  weighing  1  oz.,  1  fourth  lb.,  1  half  lb.,  1  lb.,  5  lb.,  and  10  lb. 
"With  these  and  other  objects  you  can  exercise  pupils  until  they  can 
judge  of  weight  with  considerable  accuracy. 

Original  Illustrations. — In  many  cases  you  will  interest  your 
pupils,  as  well  as  assist  them  to  comprehend  a  problem,  by  drawing 
figures  or  diagrams  upon  the  blackboard. 

Tables  of  Combinations. — Require  the  pupils  to  construct  the 
Tables  of  Addition,  Subtraction,  Multiplication,  and  Division,  before 
committing  them  to  memory.  They  can  do  this  by  using  counttrs  to 
form  the  combinations.  In  this  way  they  will  more  fully  compre- 
hend the  meaning  of  tlie  tables,  and  also  prove  them  to  be  correct. 

Only  a  small  part  of  any  table  should  be  assigned  for  one  lesson, 
and  upon  that  the  pupils  should  be  made  thorough  before  an  advanced 
lesson  is  assigned.  Let  them  from  memory  write  that  part  of  the  table 
learned  upon  their  slates  or  paper,  and  upon  the  blackboard.  Also 
yourself  write  upon  the  blackboard  portions  of  the  table,  not  in 
regular  order,  and  without  the  results,  and  require  the  pupils  to  give 
the  results  without  hesitation.  For  example,  take  the  4's  in  the  Ad- 
dition Table,  and  writing  them  as  directed, 

44:44:4:44:444 

4  8  3  9  5  0  7 


MANUAL     FOR     TEACHERS.  211 

point  to  the  numbers  forming  any  combination,  ai:!d  require  instant 
answers.  Tliis  may  be  done  first  with  the  whole  class,  and  then  with 
each  pupil  separately.  Pursue  a  similar  course  with  review  exercises 
in  all  the  tables.  Also,  give  numerous  problems  to  lae  class,  and 
require  instantaneous  answers ;  and  let  pupils  give  original  problems 
to  each  other  for  mental  solution. 

Inductive  and  Oral  Exercises. — '*  Make  haste  slowly"  in  pass- 
ing over  these  portions  of  the  book.  Exercise  pupils  upon  the 
definitions  and  signs  until  they  are  perfectly  familiar  with  them.  The 
Oral  Exercises  should  be  assigned  to  the  class  in  connection  with  the 
Tables  of  Combinations.  For  example,  when  the  class  have  learned  the 
addition  of  3's  from  the  table,  assign  the  oral  exercise  in  counting  by 
3's.  The  forms  given  for  the  first  two  or  three  exercises  under  each 
number  should  be  continued  through  all  the  exercises.  Give  your 
pupils  frequent  practice  upon  these  exercises.  Let  the  school  to- 
gether recite  from  them  5  minutes  at  a  time,  or  when  they  are  passing 
into  and  out  of  the  room,  or  when  classes  are  moving. 

All  the  possible  combinations  of  the  numbers  used  are  given  in  the 
oral  exercises.  If  pupils  are  ready  in  these,  they  will  be  ready  and 
accurate  in  computations. 

Conducting  Recitations. — In  conducting  recitations,  never  for- 
get that  the  ends  to  be  accomplished  are  fourfold,  viz. : 

1st.  To  impart  new  and  valitable  instruction,  adapted  in  kind  and 
amount  to  the  condition  of  the  minds  of  your  pupils ; 

2d.  To  teach  pupils  to  think,  by  so  guiding  their  inquiries  that  they 
shall  discover  truths  for  themselves ; 

3d.  To  make  them  thorough,  by  always  requiring  accurate  recitations 
and  explanations ;  and, 

4:th.  To  keep  them  interested  in  their  studies. 

The  following  order  in  conducting  recitations  has  been  found  to 
secure  these  results : 

1st.  Hear  as  many  of  the  class  recite  the  lesson  assigned  as  time  will 
permit,  requiring  them  to  go  through  the  recitation  without  inter- 
ruption from  other  members  of  the  class,  and  with  as  little  prompting 
and  as  few  questions  as  possible  from  you.  Throw  no  stumbling- 
blocks  in  their  way  at  this  time ;  for  pupils  who  recite  a  new  lesson 
well,  do  all  you  have  a  right  to  ask  of  them  at  first. 

2d.  After  this,  test  their  knowledge  of  the  lesson,  by  fair  but  criti- 
cal questions.    In  this  way  you  will  find  what  instruction  they  need. 

Sd.  Impart  the  needed  instruction  and  no  more,  always  observing 
this  rule :  ^^  Never  tell  a  child  anything  you  wish  Mm  to  remember,  with- 
out requiring  him  to  tell  it  to  you  again.'''' 

4:th.  Make  practical  applications  of  the  lesson. 

5th.  Review  such  portions  of  previous  lessons  as  you  deem  im- 
portant. 


212  MANUAL     FOB     TEACHERS. 

To  express  in  words  what  we  have  learned,  is  to  make  the  knowl- 
edge our  own,  to  make  it  more  clear  to  our  understanding,  and  to 
fix  it  in  our  memory.  Therefore,  require  pupils  to  write  out  upon 
their  slates  or  paper  a  full  explanation  of  one  or  more  of  the  problems 
in  the  lesson,  before  coming  to  recitation,  and  also  upon  the  black- 
board at  recitation.  The  explanations  of  examples  solved  in  this  book 
may  be  taken  as  models ;  but  as  many  of  these,  especially  in  integers, 
are  more  minute  than  you  should  require  for  the  solution  of  the 
problems,  a  few  hints  are  here  given  that  may  aid  you  in  properly 
directing  your  pupils  in  their  explanations. 

Addition. — Suppose  you  have  assigned  a  solution  and  explanation 
of  problem  81,  page  29,  to  be  written  out  by  a  pupil.  His  explanation 
may  be  as  follows:. 

i%e  farmer  raised  as  many  'bushels  of  grain  as  the  sum  of  687  hushels^ 
1,229  bushels,  643  bushels,  184  bushels,  259  bushels,  and  296  bushels. 

Since  only  figures  occupying  the  same  place  in  different  numbers  can  be 
added,  I  wrote  the  given  numbers  (or  the  given  parts)  with  ones  undfr 
ones,  tens  under  tens,  and  so  on. 

I  then  commenced  with  the  ones,  and  added  each  column  separately, 
writing  the  right-hand  figure  of  each  sum  under  the  column  added,  and 
adding  the  left-hand  figure  with  the  next  column. 

The  sum,  3,198,  is  the  total  number  of  bushels  of  grain  which  the  farmer 
raised. 

Subtraction.  —  For  example,  take  problem  57,  page  41.  The 
pupil's  explanation  may  be  this : 

The  hay  weighed  as  much  as  the  difference  between  2,656  pounds  and 
^^^  pounds. 

Since  only  figures  occupying  the  sam£  place  in  different  numbers  cam.  be 
subtracted  the  one  from  the  other,  I  wrote  the  subtrahend  below  the  min/uend, 
placing  ones  under  ones,  tens  under  tens,  and  so  on. 

I  then  commenced  with  the  ones,  and  subtracted  each  figure  of  the  subtra- 
hend from  the  corresponding  figure  of  the  minuend,  and  wrote  the  result 
directly  below  in  the  remainder. 

The  remainder,  1,669,  is  the  number  of  pounds  of  hay  in  the  had. 

You  may  require  the  pupil  to  embrace,  in  his  explanation,  the  pro- 
cess when  any  figure  of  the  subtrahend  exceeds  the  corresponding 
figure  of  the  minuend,  until  you  are  sure  he  is  familiar  with  it ;  after- 
ward it  may  be  omitted. 

Multiplication. — Take  problem  107,  page  61.  The  explanation 
of  the  solution  may  be  as  follows : 

The  cost  of  manufacturing  SGO  pianos  was  360  times  as  much  as  the  cost 
of  manufacturing  1  piano. 

I  therefore  multiplied  $270,  the  cost  of  manufacturing  1  piano,  by  360, 
and  I  obtained  $97,200,  the  total  cost. 

You  may  require  the  pupil  to  explain  the  process — writing  the  fac- 
tors, multiplying,  adding  partial  products,  and  annexing  ciphers  for 


MANUAL    FOR    TEACHERS.  213 

the  final  product— until  he  is  familiar  with  all  the  stepg ;  after  which 
an  explanation  like  the  above  is  sufficient. 

Division. —  First  Form:  both  terms  like  denominations. — Take 
problem  37,  page  74. 

Since  $6  was  the  price  of  1  cord  of  wood,  for  $6,828  as  many  cords  were 
as  the  number  of  tiwM  $6  is  contained  in  $6,828,  which  is  1,138 


Second  Form  :  the  divisor  an  abstract  number. — Take  problem  51, 
page  76. 

He  received  1  twelfth  as  much  for  building  1  rod  offence  as  he  did  for 
building  12  rods. 

I  therefore  divided  $156,  the  cost  of  building  12  rods,  by  12,  and  the 
result,  $13,  is  the  cost  of  building  1  rod. 

You  may  require  the  pupil  to  explain  all  the  steps  in  the  process  of 
Division  untU  he  is  familiar  with  them ;  after  which  the  above  form 
of  explanation  is  sufficiently  full. 

The  above  five  explanations  may  be  modified  to  meet  the  conditions 
of  any  problem  in  integers,  decimals,  compound  numbers,  percentage, 
and  for  most  problems  in  Fractions  : 

Fractions. — The  two  following  explanations  may  be  of  some  aid 
in  Multiplication  and  Division  of  Fractions. 

Problem  17,  page  176. — Seven  eighths  of  any  number  is  7  times  as 
much  as  one  eighth  of  the  number;  and  one  eighth  of  any  number  is  found 
by  dividing  it  by  8. 

I  therefore  divided  300  pounds  by  8  to  find  1  eighth  of  them,  and  then  mul- 
tiplied the  quotient,  37^,  by  7,  to  find  7  times  1  eighth,  or  7  eighths  of  them. 

The  residt,  262^,  is  the  number  of  pounds  used. 

Problem  22,  page  181. — Since  $^  is  the  price  of  1  yard  of  ribbon,  for  %i 
as  many  yards  can  be  bought  as  the  number  of  times  $^  is  contained  in  $f . 

$To  is  contained  in  $|  as  many  times  as  10  times  $^,  or  $3,  is  contained 
m  10  times  $|,  or  $\° ;  and  $3  is  contained  in  $\°,  ^J"  or  2§  times. 

Therefore,  2i  yards  is  the  required  result. 

Combinations. — Rapidity  in  computation  is  an  acquirement  much 
to  be  desired  by  all.     Stimulate  pupils  to  pronounce  results 
rapidly,  and  without  naming  the  numbers  combined.    Thus,  9 

In  Addition— Insiedu^  of  allowing  them  to  say,  7  and  5  are  13,  ^ 
12  and  6  are  18,  18  and  9  are  27,  they  should  be  taught  to  pro-  7 
nounce  the  partial  results  orally,  as  they  point  to  each  figure  ~ 
added;  thus,  7, 12, 18,  27. 

In  Subtraction — The  usual  manner  is  this  :  2  from  6  leave  S406 
4 ;  3  from  10  leave  7  ;  5  and  1  are  6,  and  6  from  14  leave  8 ;  1  ^^? 
and  1  are  2,  and  2  from  3  leave  1.  Teach  pupils  to  perform  ■^*^* 
all  the  combinations  mentally,  and  to  pronounce  the  partial  results 
orally ;  thus,  looking  at  the  6  and  2,  the  pupil  says  4 ;  looking  at  the 
0  and  3,  he  says  7 ;  looking  at  the  4  and  5,  he  says  8 ;  and  looking  at 
the  3  and  1,  he  says  1. 


214  MANUAL     FOR     TEACHERS. 

In  Multiplication — Teach  him  as  each  figure  of  the  mul-         834:5 

tiplicand  is  reached,  to  pronounce  the  product  and  the      "^ 

sum;  thus,  35 ;  28,  31 ;  21,  24;  56,  58.  584:15 

In  Short  Division— Ld  him    name  only  the  quotient       11364(4 
figure  and  the  remainder ;  thus,  2  and  3  over,  8  and  1         2841 
over,  4, 1. 

Require  pupils  to  study  the  solutions  and  explanations,  and  to 
state  the  principles  upon  which  any  given  method  is  based.  No 
rule  should  be  assigned  as  a  lesson  until  the  principles  involved 
are  clearly  understood. 

Pupils  should  commit  to  memory  the  definitions,  principles,  and 
other  matter  in  Italics,  but  in  all  cases  they  should  be  required  to 
show  that  they  understand  what  they  have  memorized. 

If  you  find  that  pupils  '  work  to  get  the  answers'  given  in  the 
book,  rather  than  to  understand  and  apply  principles  and  methods, 
change  one  or  more  figures  in  the  problems,  when  you  assign  a  lesson, 
and  the  printed  answers  will  be  of  no  help  to  the  pupils. 

Many  of  the  Review  Problems  in  each  chapter  may  be  solved  in 
more  than  one  way.  Call  out  the  diflferent  methods  from  the  class, 
and  thus  stimulate  them  to  think :  and  after  the  solutions  have  been 
presented,  exercise  their  judgment  by  requiring  them  to  state  which 
of  the  solutions  presented  is  the  best,  and  why. 

We  will  now  pass  to  the  suggestions  to  which  references  are  made 
in  various  parts  of  the  book. 


Page  10.— Exercise  pupils  in  writing  and  reading  all  the  numbers 
to  100.  Require  them  to  write  the  numbers  given  in  the  Exercises, 
in  columns,  neatly,  placing  ones  under  ones,  and  tens  under  tens,  and 
to  bring  their  work  to  recitation  for  inspection. 

Page  11. — Drill  the  class  in  writing  and  reading  numbers  of  three 
figures  until  they  make  no  mistakes.  Then  require  them  to  write 
the  exercises  given  on  page  12,  first  upon  their  slates,  and  afterward 
upon  the  blackboard.  See  that  their  work  is  neatly  done,  the  figures 
well  made,  and  ones  written  under  ones,  tens  under  tens,  etc. 

Page  13.— Practice  pupils  upon  the  two  periods,  until  the  places 
and  their  names  are  familiar  to  them.  You  will  aid  them  in  this,  by 
writing  upon  the  blackboard  two  periods  of  ciphers,  and  under  these 
require  the  pupils  to  write  the  exercises,  and  also  to  name  the  place 
occupied  by  each  cipher. 

Page  15. — To  enable  pupils  to  write  and  read  numbers  readily, 
they  should  be  able  to  tell  promptly  how  many  figures  are  required 
to  express  any  given  number  less  than  1,000,000,000.  To  secure  this 
ability  on  their  part,  you  can  frequently  question  them  in  this  man- 
ner :  How  many  figures  express  ones  ?    Thousands  ?    Millions  ?  etc. 


MANUAL    FOR    TEACHERS.  215 

How  many  figures  are  required  to  express  50  thousand  ?  Ho-w  many 
to  express  50  thousand  7  hundred  ?    To  express  50  thousand  38  ?  etc. 

How  great  a  number  can  be  expressed  by  four  figures  ?  By  seven 
figures  ?    By  five  figures  ?  etc. 

How  many  figures  are  required  to  express  ten-thousands  ?  How 
many  to  express  hundreds  ?    Hundred-millions  ?  etc. 

Require  pupils  to  point  off  all  whole  numbers  iuto  periods  of  three 
figures  each,  commencing  at  the  right. 

Explain  to  them  that,  in  reading  numbers,  they  must  always  com- 
mence at  the  left,  and  read  each  period  by  itself  as  a  distinct  number, 
pronouncing  after  it  the  name  of  the  period. 

Before  assigning  the  Review  Exercises  on  page  16,  drill  the  class 
both  in  reading  and  writing  numbers  :  first,  upon  numbers  containing 
no  ciphers ;  next,  upon  numbers  containing  one  cipher, — the  cipher 
occupying  a  different  place  in  each  number ;  then  upon  numbers  con- 
taining two  ciphers  in  all  possible  places,  both  together  and  separate ; 
then  upon  numbers  containing  three  ciphers,  and  so  on. 

The  Heview  Uzercises  (page  16)  are  test  exercises ;  and  pupils  should 
be  able  to  write  all  of  them  before  passing  to  the  next  section. 

Page  17. — See  suggestions  on  Inductive  and  Oral  Exercises, 
page  211. 

Page  19.— See  suggestions  on  TaUes  of  Combinations,  page  210. 

Page  20. — See  suggestions  on  Tables  of  Combinations,  page  210. 

Page  22. — Explain  to  pupils  that,  since  we  can  only  add  figures 
occupying  the  same  place  in  different  numbers,  and  since  it  is  more 
convenient  to  have  the  figures  to  be  added  stand  in  a  column,  we 
write  the  parts,  for  convenience,  ones  under  ones,  tens  under  tens, 
etc.  Also  explain  that  we  commence  at  the  right  to  add,  not  from 
necessity, — for  we  may  commence  at  any  other  place, — ^but  because  it 
is  more  convenient. 

In  order  that  your  pupils  may  do  their  work  methodically,  require 
them  uniformly  to  commence  either  at  the  top  or  at  the  bottom  of 
columns  of  figures  to  add  them. 

Page  24. — ^Explain  the  terms  parallel  and  Tiorizontal,  bo  that  your 
pupils  will  clearly  understand  their  meaning. 

Page  25. — The  signs  for  wood-land,  meadow,  fences,  streams,  etc., 
on  this  map  are  the  conventional  signs  used  by  topographers  and 
surveyors.  They  always  mean  the  same  thing  whenever  found  upon 
a  properly  drawn  map. 

Problem  41. — Be  sure  that  the  pupil  discovers  that  there  are  two 
fences,  each  47  rods  long,  and  two  others  each  34  rods  long. 

Page  26. — ^If  more  problems  are  required,  a  large  number  may  be 
formed  from  this  table  of  railroad  distances.  Thus,  assign  for  a  lesson 


216  MANUAL     FOE     TEACHERS. 

to  find  the  distance  from  Worcester  to  each  of  the  other  places  named 
in  the  table. 

Page  37.— Require  the  pupil  to  add  an  example,  first  by  the 
method  given  on  page  34,  and  then  by  the  method  here  given.  Then 
require  him  to  compare  the  results,  and  thus  to  discover  that  the 
latter  method  is  only  an  abridgment  of  the  former. 

Page  31. —  See  suggestions    on   Inductive   and    Oral   Exercises^ 
page  211. 
Page  33. — See  suggestions  on  Tables  of  Combinations^  page  210. 
Page  33.— See  suggestions  on  Oral  Exercises^  page  211. 

Page  35. — If  Principle  II.  is  not  understood  by  your  pupils, 
lead  them  to  see  the  truth  of  it,  by  questions.  Thus :  What  is  the 
dificrence  between  $7 and  $3  ?  Between  7  boys  and  3  boys  ?  Between 
7  tens  and  3  tens  ?    Between  7  thousands  and  3  thousands  ?  etc. 

Page  37.— The  method  of  adding  10  to  both  minuend  and  subtra- 
hend before  subtracting,  when  any  figure  in  the  subtrahend  exceeds 
the  corresponding  figure  in  the  minuend,  is  thought  to  be  too  difii- 
cult  for  quite  young  beginners,  and  hence  it  is  not  introduced  into 
this  book.  Those  teachers  who  prefer  to  use  it,  will,  of  course, 
do  so. 

By  the  aid  of  objects  show  the  class  that  1  ten  and  6  ones  are  16 
ones,  and  that  7  tens  and  6  ones  are  6  tens  and  16  ones. 

Page  40.— Require  the  pupil  to  solve  this  example  first  by  the 
method  given  on  page  37,  and  then  by  the  method  here  given.  He 
will  thus  see  that  the  latter  method  is  only  a  abridged  form  of  the 
former. 

Page  41.— Require  your  pupils  uniformly  to  call  the  next  left- 
hand  figure  of  the  minuend  1  less,  or  the  next  left-hand  figure  of  the 
subtrahend  1  more. 

Page  45.  —  See  suggestions   on   Indtcctive   and    Oral   Exercises, 
page  211. 
Page  47. — See  suggestions  on  Tables  of  Combinations,  page  210. 

Page  48. —  See  suggestions  on  Indicctive  and  Oral  Exercises, 
page  211. 

Page  49. — Very  few  learners  fully  comprehend  the  fact  that  every 
problem  in  multiplication  can  be  solved  by  addition.  In  order  to  fix 
this  fact  firmly  in  the  minds  of  your  pupils,  require  them  to  solve  the 
first  10  problems  on  page  50,  both  by  addition  and  multiplication. 

Page  53, — Require  pupils  to  speak  of  the  true  multiplicand  only 
as  the  multiplicand.  For  example,  in  finding  345  times  7,  they  may 
multiply  345  by  7 ;  but  in  the  explanation  of  their  solution,  require 
them  to  say  345  times  7. 


67000 
2100 

67000 
21 

6700O 
134000 

67 
134 

140700000 

1407000 

lOO 

MANUAL    FOR    TEACHERS.  217 

Page  61,— If  this  explana- 
tion is  not  fully  understood, 
solve  the  example  in  diiferent 
ways,  and  compare  the  results. 
Thus,  you  may  use  either  of  the 
foUowing  solutions :  140700000 

But,  in  practice,  never  permit  pupils  to  write  the  factors  for  multi- 
plication as  shown  in  the  first  of  these  solutions. 

Page  62.— First  Reference.— Be  sure  to  give  your  pupils  clear 
ideas  of  the  terms /actors  and  parts.  Thus,  7,  5,  and  2  are  factors  of 
70,  but  they  are  parts  of  7  +  5  +  2,  or  14,  and  also  of  752. 

Second  Reference.- Round  trip,  out  and  back  again,  or  twice 
over  the  road,  once  each  way.    This  is  also  called  doubling  the  road. 

Third  Reference. — See  suggestions  on  Conducting  JRecitations, 
page  211. 

Page  64.— See  suggestions  on  Inductive  Exercises^  page  211. 

Page  65. — Illustrate,  by  the  use  of  objects,  fractional  division,  or 
the  division  of  a  given  number  of  things  into  a  certain  number  of 
equal  parts.  For  example,  with  a  quantity  of  beans  or  com,  or  a  large 
number  of  other  counters  before  the  class,  divide  the  number  of  objects 
into  2, 3,  4,  5,  etc.,  equal  parts,  and  then  require  each  pupil  to  do  the 
same.  Also,  ask  numerous  questions  like  these  :  How  can  you 
find  1  half  of  these  objects  ?  How  1  third  of  them  ?  How  1  fifth  of 
them  ?  1  fifteenth  ?  1  fortieth  ?  etc.  How  can  you  find  1  third  of 
any  number?  How  1  eighth  of  it  ?  How  1  twenty-fourth  ?  1  seventy- 
first  ?    1  ninetieth  ?  etc. 

Page  66, — See  suggestions  on  TaUes  of  Comiinations,  page  210. 

Page  67.— See  suggestions  on  Oral  Exercises,  page  211. 

Page  73. — Require  the  class  now  to  solve  the  problems  on  pages 
70,  71,  72,  and  73,  by  short  division. 

Page  75.— Do  not  permit  pupils  to  leave  these  12684  l  28 

principles  until  they  thoroughly  comprehend  them.  1 4413 

You  can  test  their  understanding  by  placing  erro-  ^^o 

neons  processes  upon  the  blackboard,  and  requiring  -^g 

them  to  point  out  the  errors,  and  to  tell  why  they  28 

are  wrong.    Thus,  place  upon  the  blackboard  this  84 

example,  and  require  pupils  to  correct  it,  and  to  ^ 
give  reasons  for  the  correction : 

Page  98. — If  your  pupils  are  slow  to  learn  the  notation  of  deci- 
mals, you  may  find  the  following  direction  of  value  to  them : 

To  Wrue  any  Decimal  Number, 

Write  it  first  as  an  integer,  and  then  place  ihe  deeimdl  poirU  according 
to  the  table  of  values  of  decimal  numbers. 

R 


218  MANUAL    FOR    TEACHERS. 

Exercise  your  pupils  until  they  are  familiar  with  this  table.  Ask 
such  questions  as  the  following,  until  they  can  give  correct  answers 
promptly  : 

How  many  decimal  figures  express  thousandths  ?  How  many  hun- 
dred-thousandths ?    Hundredths?    Ten-millionths ?  etc. 

"What  decimal  is  expressed  by  four  decimal  figures  ?  By  three 
figures  ?    By  eight  figures  ?  etc. 

Page  99. — Pupils  should  be  thoroughly  grounded  in  this  table,  as 
a  knowledge  of  it  will  enable  them  to  master  the  theory  of  computa- 
tions in  decimals  with  increased  facility.  For  a  test  of  their  knowl- 
edge, write  a  line  of  I's  on  the  blackboard, 

111111111.11111111, 
and,  commencing  at  the  right,  require  your  pupils  to  give  the  law  of 
increase  toward  the  left,  first  to  the  decimal  point,  and  afterward  to 
hundred-millions ;  thus,  10  hundred-millionths  are  1  ten-millionth, 
10  ten-millionths  are  1  millionth,  10  millionths  are  1  hundred-thou- 
sandth, and  so  on,  to  10  ten-millions  are  1  hundred-million. 

Then,  commencing  at  the  left-hand  figure  of  the  integer,  require 
them  to  give  the  law  of  decrease  to  the  right ;  thus :  1  hundred-mil- 
lion is  10  ten-millions,  1  ten-million  is  10  millions,  1  million  is  10  hun- 
dred-thousands,   1  one  is  10  tenths,  1  tenth  is  10  hundredths,  1 

hundredth  is  10  thousandths, 1  ten-millionth  is  10  hundred-mil- 
lionths. 

After  the  class  can  run  through  this  number  with  facility,  write 
lines  of  4's,  7's,  3's,  and  so  on,  and  practice  them  upon  the  numbers 
thus  formed  in  the  manner  above  directed. 

Page  100.— First  Reference.— Illustrate  these  principles  by 
numerous  examples  like  these;  show  that  .5  =  .50  =  .500  =  .5000; 
that  .75  =  .750  =  .75000,  and  so  on ;  also,  that  annexing  a  cipher 
changes  the  values  of  the  figures  of  an  integer,  by  removing  ones  to 
tens,  tens  to  hundreds,  etc. 

Second  Reference.— In  reading  numbers,  the  word  and  should  be 
used  only  between  the  integral  and  decimal  parts  of  mixed  numbers. 
If  441  be  read  four  hundred  and  forty-one,  certain  entirely  difier- 
cnt  numbers  would  be  read  exactly  alike.  Thus  400.041  and  .441 
would  both  be  read  four  hundred  and  forty-one  thousandths. 

The  following  unlike  numbers  would  be  read  alike :  15000,00008  and 
.15008;  5000.0024  and  .5024;  964000.000072  and  .964072. 

Ten-thousand,  hundred-thousand,  ten-thousandths,  hundred-thou- 
Bandths,  etc.,  should  be  written  as  compound  words ;  otherwise,  three 
hundred  ten  thousandths  might  mean  .310  or  .0300. 

The  exercises  on  page  100  will  test  the  ability  of  pupils  to  write 
and  read  decimals.  For  additional  practice,  write  decimal  numbers 
upon  the  blackboard  for  your  pupils  to  read,  and  require  them  to 
write  numbers  from  dictation,  until  they  can  write  and  read  decimals 
with  the  same  facility  as  integers. 


MANUAL    FOK    TEACHERS.  219 

Page  101, — ^Explain  the  difference  between  Addition  of  Decimals 
and  Addition  of  Integers.  Do  the  same  with  Subtraction,  Multipli- 
cation, and  Division. 

Page  129. — Be  sure  that  pupils  always  write  abbreviations  cor- 
rectly, and  in  their  proper  places,  i.  e.y  that  they  write  the  proper  ab- 
breviated form,  and  place  a  period  after  it ;  and  that  the  abbreviation 
be  placed  after  the  number,  except  the  sign  of  dollars  (^),  which 
should  always  be  placed  before  the  number. 

Before  your  pupils  take  up  the  tables  in  Compound  Numbers,  read 
carefully  the  suggestions  on  Use  of  Objects,  page  209, 

Page  146. — Make  your  pupils  so  familiar  with  this  table,  that 
they  can  tell  at  once  the  number  of  any  month,  and  the  number  of 
days  in  it. 

Page  147. — In  the  table  of  weight  (see  185),  the  denomination 
grain  is  not  named,  as  it  is  not  used  in  commercial  weight.  Explain 
to  your  pupils  that  if  a  pound  is  divided  into  7000  equal  jmrts,  1  of 
these  parts  is  1  grain :  and  hence,  a  grain  is  1  seven-thousandth  of  a 
pound;  a  gram  is  15432  seven-miUionths  of  a  pound;  and  a  pound 
contaitis  a  little  more  than  453.6  gram^. 

Page  165.— Thorough  drills  upon  these  principles  are  indispens- 
able to  the  further  intelligent  progress  of  the  pupil.  Therefore,  be 
sure  that  he  understands  them  before  passing  to  Section  II. 

Page  169. — The  common  solution  is  similar  to  Reduction  Ascend- 
ing, and  it  may  be  explained  in  the  same  manner,  by  regarding  the 
numerator  as  a  number  to  be  reduced  to  a  higher  denomination,  and 
the  denominator  as  that  number  of  the  given  denomination  that 
equals  one  of  the  required  higher  denomination.  Thus,  the  process 
of  reducing  17  fourths  to  a  mixed  number  is  the  same  as  that  of 
reducing  17  pecks  to  bushels,  or  17  quarts  to  gallons. 

Page  170.— The  common  solution  is  similar  to  Reduction  De- 
scending, and  may  be  explained  in  the  same  manner,  by  regarding  the 
integer  as  the  higher  denomination,  and-the  numerator  of  the  fraction 
as  the  number  of  the  next  lower  denomination.  Thus,  the  process 
of  reducing  5|  to  fourths  is  the  same  as  that  of  reducing  5  bu,  3  pk. 
to  pk.,  or  5  gal,  3  qt,  to  qt. 

Page  173.— Fix  in  the  minds  of  pupils  the  fact  that  numerators 
only  are  added  or  subtracted,  and  denominators  are  written  to  give 
name  or  denomination,  the  same  as  in  compound  numbers.  Give 
numerous  illustrations,  such  as  4  men  +  3  men  +  5  men ;  $4  +  $3  + 
$5;  4  parts  +  3  parts  +  5  parts  ;   4  ninths   +  3  ninths  +  5  ninths,  or 

£i^s  •*•  ^!ths  +  ^s'  1^  P^°^-S  P^^«5  S13-$8;  13  parts- 

13  8 

8  parts ;  13  ninths  —  8  ninths,  or  —. — ,  etc. 

ninths       ninths 


220  MANUAL    FOR    TEACHERS. 

Page  182.— The  method  "Invert  the  divisor  and  proceed  as  in 
multiplication,"  is  introduced  in  Cancellation,  p.  184.  Therefore 
pupils  should  be  required  to  solve  and  explain  these  problems  in  the 
manner  here  shown,  as  this  course  wiU  make  them  familiar  with  the 
reasons  for  the  process,  which  is  not  likely  to  be  the  case  if  they  at 
once  adopt  the  mechanical  convenience  of  inverting  the  divisor. 

Page  184.— Be  sure  that  pupils  understand  that  a  factor  in  either 
term  will  cancel  only  one  like  factor  in  the  other  term.  Also  alwavs 
require  them  to  write  1  in  the  place  of  a  canceled  term,  as  pupils 
are  liable  to  think  that  when  a  term  is  canceled,  a  0  belongs  in  its 
place.  Test  their  knowledge  upon  this  point,  by  questions  and  ex- 
amples. 

Page  19T.-Exp]ain  the  fact  that  any  number  of  months  forms 
the  numerator  of  a  fraction,  of  which  12  (the  number  of  months  in  a 
year)  is  the  denominator;  and  that  this  multiplying  fraction  can  often 
be  reduced  to  lower  terms  before  multiplying.  Thus,  2  mo.  =  /-  =  i 
yr.,  3^ mo.  =  xV  -  i  yr.,  4  mo.  =  ^\  =  \  yr.,  6  mo.  =  r%  =  i  yr.,  8  mo". 
-  /a  -  i  yr.,  9  mo.  =  t\  =  f  yr.,  10  mo.  =  ia  =  s  yj.^  ^  familiarity  with 
this  fact  will  often  enable  pupils  to  abridge  their  work  by  multiply- 
ing the  interest  for  1  year  by  the  time  expressed  in  years  and  frac 
tions  or  decimals  of  a  year. 

Page  201. — When  pupils  pass  over  these  General  Review  Prob- 
lems, exercise  their  ingenuity  in  producing  diflferent  methods  for 
solving  the  same  problem.  Thus,  problem  30,  page  203,  may  be 
solved  in  three  ways  : 

1st  Find  the  contents  of  the  pile  in  cubic  inches,  and  then  divide 
them  by  the  number  of  cubic  inches  in  a  brick. 

2d.  Find  the  cubic  contents  of  the  pile  in  feet,  and  multiply  them 
by  the  number  of  bricks  in  a  foot ;  and, 

M.  Find  how  many  bricks  long,  wide,  and  high,  the  pile  is,  and  then 
multiply  these  numbers  togetjier.    Again, 

Problem  50,  page  104,  may  be  solved  in  two  ways : 

Is^.  Find  the  whole  number  of  pounds  raised,  by  multiplying  the 
number  of  pounds  per  acre  by  8.5,  the  number  of  acres,  and  multiply 
$.06,  the  price  of  1  pound,  by  this  product ;  and, 

2d.  Find  what  he  received  for  1  acre  of  flax,  by  multiplying  $.06, 
the  price  per  pound,  by  850,  the  number  of  pounds  raised  to  the 
acre ;  and  then  multiply  this  result  by  8.5,  the  number  of  acres  raised. 

This  course  will  teach  pupils  that,  in  many  cases,  there  is  more  than 
one  right  way  to  solve  a  problem,  and  it  will  also  teach  them  to 
think  methodically.  ^  ^ 


YB    17389 


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